Physics - Introduction To String Theory

Lecture 1

Quantum Field Theories: An introduction 
 



The string theory is a special case of a quantum field theory (QFT). Any QFT deals of Riemannian manifolds, the dimension of is with smooth maps the dimension of the theory. We also have an action function defined on the set Map of smooth maps. A QFT studies integrals

 





$ % '& )(+* &-, (1.1)  !#" Here (+* &-, stands for some measure on the space of paths, . is a parameter (usually %
Map  / 021 is an insertion function. The very small, Planck constant) and 657<9=8/: should number as the probability amplitude of the contribution ;
 tobetheinterpreted  4  3 of the map integral. The integral >0?A@  $ED & (1.2) 4    BC is called the partition function of the theory. In a relativistic QFT, the space  has a Lorentzian metric of signature GF #HI KJKJJ4 /H . The first coordinate is reserved for time, the rest are for space. In this case, the integral (1.1) is replaced with > @  65798/: % M& G(;* &N, J (1.3) 4  7L 3 Let us start with a O -dimensional theory. In this case  is a point, so &
P is a point QSR  and 
<TU1 is a scalar function. The Minkowski partition function of the theory is an integral >V@  WB98/: D Q J (1.4)  7L 3 Map

Map

Map

Following the Harvard lectures of C. Vafa in 1999, let us consider the following example: 1

2

LECTURE 1. QUANTUM FIELD THEORIES: AN INTRODUCTION

X =  [ e  [ XY 'Z @ \^])_ 4`Kba D ] @dc \f]Gg/_ 4`Kbaih D ] J (1.5) This integral is convergent for Re MZ kj O but can be meromorphically extended to the whole plane with poles at ZlRSmn \ . We have XY 'Z HdoB @ ZBXY MZ p XY oB @ o q Xr co @ts u J @ s v ]Gw in (1.5), we obtain the Gauss integral: By substituting ]  [ D @ XY @zy u J (1.6) [  4x/aih ] v ` 8g` g v v  Although in the substitution above is a positive real number, one can show that v v formula (1.6) make sense, as a Riemann integral, for any complex with Re 0{ O . v When Re Ej O this is easy to see using the Hankel representation of XY 'Z as a v contour integral in the complex plane. When is a pure imaginary, it is more delicate and we refer to [Kratzer-Franz], 1.6.1.2. v @|u , we can use Taking D~} @ D ] ~€‚aih to define a probability measure on 1 . It is called the Gaussian measure. Let us compute the integral >  9ƒ @  [ WB† D} @  [ W WB† D Q J [ 7L…„ [  € h)‡ L6„ @ o7ˆ . We have   Here ƒ > 9ƒ @  [ )F u Q g Ž [ i’“ƒ Qb”  ˆ‚•—– D Q J ‘ \ [Š‰p‹Œ  Obviously, [ Q g™˜ `  š€ W h D Q @ O J [  Also c ›ŠF oB @ [ o ž o B  ž Ÿ K     @ u @ D H ™

ˆ W Q Q XY 9› c ˜ ‡œ c‚u ˜ [ g ˜  € h h  Example 1.1. Recall the integral expression for the -function:

3

c u c › #›– – @V  c › ™ˆ c‘u ˜

˜ g ˜ where   c › @ c c › #–– @ o –¡ c c;› ¢  ¡ c ›£c F c ¢ K ¡ cc7¢ ˜› › c is equal to the number of ways to arrange › objects in pairs. This gives us > Mƒ @ orH Ž [ GF o7  ƒ g    M¤ •¥ ˆ c •¥ #– c‘u ”  J (1.7) ‘ ` Observe that to arrange ¤ • objects in pairs is the same as to make a labelled 3-valent c graph X with • vertices by connecting 1-valent vertices of the following disconnected graph:

b 2n

b2

b1

c2

c1

c

a2

a1

2n

a 2n

Fig. 1 This graph comes with labeling of each vertex and an ordering of the three edges emanating from the vertex. Let be such a graph, be the number of its vertices and be the number of its edges. We have , so that for some . Let

c •— ¦§¦§ M MX X @ Ÿ~•

Then

•

X

% Ÿ%

9 9X X ¨ @ c §¦ MX

© 9X @ ) Fž% ’ªƒ « # !– ¬ c‚u o ? !¬ J MX > Mƒ @ orH Ž © MX p

¬

% 9X @

e­ 9® ­A 9®

where the sum is taken over the set of labeled trivalent graphs. Let be the number of labelled trivalent graphs which define the same unlabelled graph when we forget , where is the number of about the labelling. We can write labelling of the same unlabelled 3-valent graph . Thus

© ¯ M® @ A­ M® © 9X ® > 9ƒ @ o°H Žp± © ¯ 9® #

where the sum is taken with respect to the set of all unlabelled 3-valent graphs. It is easy to see that

c p– Ÿ-–² 

­A 9® @ ³ ¥• Aut M® g

4

LECTURE 1. QUANTUM FIELD THEORIES: AN INTRODUCTION

±

© ¯ M® @ Gc Fž¥• ’“ #ƒ – c‚ g u  c  •¥³ #– Ÿ-–² g  @ c‚u ) F? ¤‘± ’“ ƒ ³ «   J ” Autc 9® Aut M® • Given an unlabelled 3-valent graph with vertices, we assign to each vertex a factor c‘u , then multiply number GF¤~’ , toofeach edge a factor o7ˆ all the factors and divide by the symmetries of the graph. This gives the Feynman rules to compute the contribution of this graph to the coefficient at ƒ . For example, the graph g contributes )F¤‘’   † g g €` ` ` g @ F´µ €” † and the graph @ F·¶ † J The total coefficient at ƒ is F † . This contributes )F¤‘’ g   † µ g €` ` at ƒ g in`G¸ > € 9ƒ given by the formula (1.7).g `G¸ `)€¹ coincides with the coefficient Recall that the Principle of Stationary Phase says that the main contributions to the integral ~º WB D 7L6»B¼ &½ iQ Q when ¾ goes to infinity comes from integrating over the union of small comapct neighborhoods of critical points of ¿À iQ . More precisely we have the following lemma: % Lemma 1.1. Assume &½ 9Q has a compact support and ¿À iQ has no critical points % • on . Then, for any natural number , ÁÂ6à [ ¾   [ WB &Å 9Q D Q @ O J »7ĕ  [  L6»B¼ @ O . Integrating by parts, Proof. We use induction on . The assertion is obvious for • we get ’  [ WB7Æ &Å 9Q w D Q @ ’ Æ &½ iQ WBÈÈ [ [ H  [ WB &Å 9Q D Q @ ¾ [  L6»B¼ ¿À 9Q w#Ç ¾ ¿À 9Q w/Ç  L»7¼ È [  L6»B¼    [ 6W D J [ BL6»7¼ &½ iQ Q  , we get Multiplying both sides by ¾ ‡Å` ÁÂ6à [ ¾   [ WB &Å 9Q D Q @ ’ Á6Âà [ ¾   [ W7Æ &½ iQ w D Q J ž iQ w#Ç ½ ‡ `   6 L B » ¼  L 7 » ¼ [ [ »7Ä »BÄ   Applying the induction to the function Æ 56WWBi É w we get the assertion. ¼ Ç so that

5

¿À iQ Q KJJKJ4 QÊ

¿ 9Q Ë Q `

Í Ì i Q ½ & i Q L L L •Îj O Ð  [ 6W98/: D @ Ž <Ï°Ð WB98/: D HeÑ  pJ &½ 9Q Q 7L6¼ &½ iQ Q M. [ 7L…¼ L  @ D HVo , where D Now let us consider a QFT in dimension 1. Usually we write ( o HV@ o is the O is the space-dimension, and is the time-dimension. A QFT in dimension @  1 * K

o quantum mechanics. In this case, we take to be equal to , Ò O , or  ` 1—ˆ m Ó
< Ô  Õ    parametrized by ] . A map is path in (infinite, or finite, or a loop). The action is defined by  @ )

ž ] ×Ö ] # šØ ] ) D ]

where
ÙfÚ1 is a smooth function defined on the tangent space of  (a LaÖ The expression Ù  ] p šØ ] ) D ] is a density on  equal to the composition grangian). D ;
‘ Û Ù =  ¨  Ö of the differential and . @ 1  so that @ 1 ÝÜ 1  with coordinates 9Þ Þ Ø . For  Ù  Ö For example, take   v any 9Þ Þ Ø and a map ß
š* à , á1 , 9ß° ] p ßYØ ] ) is obtained by replacing Þ with Ø Ø

Ö ß° ] Aandcritical Þ withpointßY ] of. the functional ž iß°Ö ) satisfies the Euler-Lagrange equation â p Ø ) @ ] D â # Ø ) #J âãÖ iß° ] ßY ] D ] â ãÖ Ø 9ß° ] ßr ] (1.8) L For example, let us take theL Lagrangian Ž  ã Ø F % ã KJJKJ 㠏 (1.9)  Lg ` L` Then we get from (1.8) › D g D QÅ] ] @ F0ä % 9QÅ ] ™ #J g ¿À 9Q

has finitely many critical points , we write our function Thus if as a sum of functions with support on a compact neighborhood of and a function which has no critical points on the support of and obtain, for any ,

Thus a critical path satisfies the Newton Law; it gives the major contribution to the partition function. Fix and . Let be the space of smooth maps such that . The integral

 +
PQ  Q w R

] @]Gw Q R G  @ Q åæ ] Q¥ç ]Gw Q w ] ]Gw è w (1.10) Ëe ] Q¥ç ] w Q w @   W‚é Ʌ WBÉ!  L 3 !ê a iM8 : (;* ] , a a that a particle in the position Q at the can be interpreted as the “probability amplitude” moment of time ] moves to the position Q w at the time ]Gw .

6

LECTURE 1. QUANTUM FIELD THEORIES: AN INTRODUCTION

 @ 1

%



åæ ] QÅç ] w Q w * ] ]Gw , @ ­ åæ ] QÅç ]Gw Q w @ ƒ @ ]Gw F ] ™ˆ ­ ­ @

] KJJKJ ]G ë ]Gë Q ] ] G ] w @  ì
Û* ] ]Gw , q1 ` Q Q g KJKJJ4 Q ë Q ë` Q g w * 1 , ‡Å` ‡Å` ] L ] L ‡½` ] @ Q H Q ] L ‡Å` FEF Q] L ] F ] #J L L L ‡½` L  L It is clear that the set of such paths is bijective with 1 ë integrate ` and soî weë . can a function Ì
åæ ] Q¥ç ]Gw Q w lí1 over this space to get a number Now we can define (1.10) as the limit of integrals î ë when ­ goes to infinity. However, this limit may not exist. One of the reasons could be that î ë contains a factor ï ë for some constant ï with ð ïÍð jño . Then we can get the limit by redefining î ë , replacing it  with ï ë î ë . This really means that we redefine the standard measure on 1 ë    D D  the measure Q on 1 by ï 4` Q . This is exactly what we are1  going 4to` replacing

do. Also, when we restrict the functional to the finite-dimensional space É À N ëØ D ]` byof piecewise linear paths, we shall allow ourselves to replace the integral ò a Ö side its Riemann sum. The result of this approximation is by definition the a right-hand in (1.10). We should immediately warn the reader that the described method of giving

Let us compute it for the action defined by the Lagrangian (1.9) with . We shall assume that the potential function is equal to zero. The space is of course infinite-dimensional and the integration over such a space has to be defined. Let us first restrict ourselves to some special finitedimensional subspaces of . Fix a positive integer and subdivide the time interval into equal parts of length by inserting intermediate points . Let us choose some points in and consider the path such that its restriction to each interval is the linear function

a value to the path integral is not the only possible. We have

 [ JKJKJ  [ * ’ª› ëŽ Á  6  à @



c ƒ  iQ FÎQ g ,óï  ë D Q g JKJKJ D Q ë J Ë= ] Q¥ç ] w Q w ë [ Ä  [  [ ‰p‹Œ L ` L L ‡Å` (1.11)   D Here Q KJKJJ4 Q ë are vectors in 1 and Q is the standard measure in 1 . The number ï shouldg be chosen to guarantee convergence L in (1.11). Using (1.6) we have [ [ 6W W  W W †  D Q @  g Æ W † Ç h W W †  D Q @ [ x œ  h h#4x h  h g [   x h kô œGõ h ô köh œ  h g   [  @ W W †  Wg D Q @ y c u v W W †  J   öh œ  h [   x h   öh œ  h  Next [ * v F c 9Q FÎQ ” g F v iQ ” FÎQN÷ g , D Q ” @ [£‰p‹Œ ` 

7

@  [ * F Ÿ c v Æ Q F Q H Ÿ Qb÷ g F vŸ iQ FEQb÷ g , D Q @ y c‚Ÿ uv † W Wù™ J ” ` Ç ”  ö œ h [ø‰K‹Œ `  Thus [ * v F iQ FÎQ g g F v 9Q g FEQ ” g F v iQ ” FÎQ ÷ g , D Q g @ [ú‰K‹<Œ `  y c u v y c‚Ÿ uv * F vŸ iQ FEQb÷ g , @ y Ÿ u v g * F vŸ iQ FEQb÷ g , J ‰K‹<Œ ` g ‰p‹Œ ` Continuing in this way, we find u  [ * v ëŽ v F  iQ FÎQ g , D Q g JJKJ D Q ë @ y ­ v ë ë 4` * F ­ 9Q FÎQ ë g ,

[Š‰K‹Œ ‰K‹<Œ ` ‡Å` L L ‡Å` 4  ` ` L  v @ ˆc J where › ’“ƒ If we choose the Ðconstant ï equal to ï @¨û ˜ Ð hœ then we will g € L6„ü be able to rewrite (1.11) in the form Ë= ] QÅç ] w Q w @ýû c‚u › ’“­+ƒ hœ þ  ô Éiÿ ô " h @úû c‚u ’# ›] F ] hœ  þ  ô É É9ÿÿ ô " h J (1.12) w ü h " ü h We shall use Ëe ] Q¥ç ]Gw Q w to define a certain Hermitian operator in the Hilbert D} the space that for any manifold  with some Lebesgue measure  1— . D}Recall g

Ö space g of square integrable complex valued functions modulo funcÖ to zeroconsists tions equal on the complement of a measure zero set. The hermitian inner product is defined by

¿ @  ¯¿ D~} J  D~} is a Hilbert-Schmidt operator: Example 1.2. An example of an operator in g 

Ö Ù ¿À iQ @  Ëe 9Q - ¿À - D}

 Ü Ü } }   is the kernel of Ù . In this formuladwe integrate where Ë= iQ - R g Ö Ë= iQ - keeping Q fixed. By Fubini’s theorem, for almost all Q , the function }is -integrable. This implies that Ù M¿ is well-defined. Using the Cauchy-Schwarz inequality, one can easily checks that    ð6ð Ù ¿—ðð g @  ð Ù ¿—ð g D~} ð6𠿗ðð g   ð Ëe 9Q  ð g D~}D}






























8

Ù

i.e.,

LECTURE 1. QUANTUM FIELD THEORIES: AN INTRODUCTION is bounded, and

6ð ð Ù ðð g @  \ ð6ð 6ð ٠𠿗¿—ð6ð ðð g Œ g ¼ We have    )Ù ¿ @  û  ¿À - Ëe iQ - D} ü 











 

- D}D~} J   ð Ë= iQ ð g   iQ D} @   Ë= iQ - ¿À - 9Q D}D} J 









This shows that the Hilbert-Schmidt operator is self-adjoint if and only if

Ë= iQ - @ Ë=  Q Ü . outside a subset of measure zero in 





In quantum mechanics one often deals with unbounded operators which are defined only on a dense subspace of a complete separable Hilbert space  . So let us extend  the notion of a linear operator by admitting linear maps where is a dense linear subspace of  (note the analogy with rational maps in algebraic geometry). For such operators we can define the adjoint operator as follows. Let denote the domain of definition of . The adjoint operator  will be defined on the set

( 

Ù

Ù

R ( Ù 

Ù

( Ù @ 

( ( Ù



R
\  W k š ð Ù ð6 iðQ QÀ# ð ð ð Œ 

( Ù Ù i Q # @ Q

( Ù Ù











 

! 



J

Q  Ù i Q #

Ù ( ÙÛ @ ( Ù K 



 Take . Since is dense in  the linear functional extends to a unique bounded linear functional on  . Thus there exists a unique vector  $"% "

 # such that . We take " for the value of  at . Note that & & is not necessary dense in  . We say that is self-adjoint if  and . We shall always assume that cannot be extended to a linear operator on a larger set than . Notice that cannot be bounded on since otherwise we can extend it to the whole  by continuity. On the other hand, a self-adjoint operator is always bounded. For this reason self-adjoint linear operators with   (' are called unbounded linear operators. 

(

R Ù Ù p @ Ù

Ùú
@  ( ÙÛ

Ù

Ù

( ÛÙ

Ù

@ g 1 D Q and define the operator Ö Ù ¿ @ ’G¿ w @ ’ DD ¿ Q J

Example 1.3. Let us consider the space 

Obviously it is defined on the space of differentiable functions with square integrable derivative. This space contains the subspace of smooth functions with compact support which is known to be dense in . Let us show that the operator  is self-adjoint. Let . Since ,

Ùt
~(Š 1 D Q 1k D g (  Ù

Ö ¿;R ¿wR Ö g Q  D @  ¿\ a w 9Q ¿À iQ Q ð ¿À ] ð g F|ð ¿À 9O ð g F \ a À¿ 9Q ¿ w 9Q D Q

9

Á6Âà [ ¿À ] ð ¿À iQ ð g ] GF / H ] aÄ ¿ R ( Ù ¿  @ Á Â6à [  \ a ’G¿ w 9Q 9 Q D Q @ Á6Âà [ û ’G¿À ] 9Q ÈÈ ‡ [[ F  \ a ’G¿À iQ w 9Q D Q @ È ü aÄ a Ä   @ ÁÂ6à [ \ a ¿À iQ ’ iQ D Q @ M¿ ™Ù  pJ w @ ( Ù  aÄ This shows that ( t( Ù K and Ù is equal to Ù on ( . The proof that ( is more subtle and we omit it.

be two copies of the space g  D} . Let Ù  É be the Hilbert-Schmidt Let

]Gw as real parameters: operator `  g defined by a kernel Ë= Ö ] Q¥ç ]Gw Q w which has a a ] ` g Ù '& iQ @  Ë= QÅç Q &½ iQ D~} J É  ] ]w w w  aa

* , we see that exists. Since is defined for all . Letting go to )  is integrable over , this implies that this limit is equal to zero. Now, for any  

 , we have











-,

.



+





.

.

.

Suppose our kernel has the following properties:

 É Ëe ] QÅç ] w Q w²w @ a  Ë= ] Q¥ç ] w Q w Ëe ] w Q w ç ] w Q w D~}  D ] w ] ] w ç a (N)  ð Ë= ] Q¥ç ] w Q w ð g D}  @ o ç (T) Ë= ] Q¥ç ] w Q w @ Ë= ] g Q¥ç ] wg Q w if ] wg F ] g @ ] w F ] ç `  ` D~} , the function ` ` (C) for any & R gÖ ] ] @  9Q Ëe ] Q¥ç ] w Q w &½ 9Q w D}  D~} Â6à 6 Á is continuous for ]Gw j ] and É ] @ 0 & pJ a Ä property ai‡ e
Å* (M) When Ë is defined by the path integral, as one of the axioms of

] ]Gw , is2taken  from QFT. It expresses the property that any path
¥* ] ]Gw ,   from Q to Q w and a path
¥* ]Gw ]Gw ,Q 2to Q w isfromequalQ w toto a sum of paths Q w . Property (N)` says that the total probability amplitude ofg a particle to move from Q to somewhere is equal to 1. Notice that property (N) implies that the operator Ù  É is unitary. In fact, aa  Ù KÙ D} @  a a É & a a É  (M)



/

10

32

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0

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LECTURE 1. QUANTUM FIELD THEORIES: AN INTRODUCTION

10

 û  D} û  D} D~} @   Ë= ] QÅç ] w Q w &Å 9Q  ü  Ëe ] Q¥ç ] w Q w iQ  ü   û  D}  D~} J @  D }

 &½ 9Q  Ëe ] QÅç ] w Q w Ëe ] Q¥ç ] w Q w  ü iQ   &Å 9Q 9Q  Now we use the following Stone-von Neumann’s Theorem: Theorem 1.1. Let ] p ] R 1 \ be a family of unitary operators in a Hilbert space . Assume that @ Í ]  is continious for ] j O and 

Í Ì (i) for ] ] Á6Âà all\ ÌÍR ] @ , the function  ; Ä ] ‡ ]Gw R 1 \ ] H ]Gw @ ] ]Gw pJ (ii) for aall Then ( @ SR
ÁÂ6à \ Í ] FÓÒ exists ] is dense in and the operator defineda Ä by‡ @ ’ ÁÂ6à \ ] FÓÒ aÄ ‡ ] is self-adjoint. It satisfies ] @ 7L a ] { O J Applying this to our situation, we obtain that Ù  @   ] { ]\  L aªba is called the Hamiltonian operator assoa a for some linear operator . The operator ciated to Ëe ] Q¥ç ]Gw Q w . We would like to apply the above to our function Ëe ] QÅç ] w Q w @ýû c‚u ’# ›] F ] hœ Æ ’'›Óc i Q ] w F FÎ] Q g J w ü ‰p‹Œ w Ç Unfortunately we cannot take the function Ëe ] Q¥ç ]Gw Q w to be the kernel of a HilbertD D Schmidt operator. Indeed, it does not belong to the space g 1 g Q Q w . In particular Ö (T) is obviously true and property (N) is not satisfied. One can show that (M) is OK,

(C) is true if one restricts to functions & from a certain dense subspace of g 1— . Ö The way about this is as follows (see [Rauch]). 1 First let us recall the notion of the Fourier transform in . It is a linear operator defined on the Schwartz space Û 1— 1—  of smooth functions with all derivatives g Ö tend to zero faster than any power of ð Q½ð as Q . It is given by the formula [  M&½ iQ ) ž
@ &À @ s o c‘u  L W &½ 9Q D Q J [ Here are some of the properties of this operator: 4/

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8

9

:

;8

9

;8

6

5

>

5

8

=<

5

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49

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8



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8

@

5

A

A

.

@

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C,

B

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-E

GF

H

11


 1° 0  1° is an unitary operator; (ii) ` M&½ 9Q ) @ M&½ )FžQ ) ; @ ’ &½ ™ ; (iii) '&Å 9Q w @ GFž’ªQš&½ 9Q ) ç (iv) &À w k @ s c‚u M& , where (v) '& [ @

& 9Q &½ iQ F - - D J [  Let us show that our function Ëe ] Q¥ç ]Gw Q w is the propagator for the Schr¨odinger equation â ⠒ â ] H › c â Q g ½ ] Q k
@ ’ H › c WBW @ O ½ MO Q @ ¿À 9Q R Ö g 1— #J g @ o . Supose a ¿À 9Q R Û 1° . Let us find the solution in We take for simplicity ›  1— using the Fourier transform. Using property (iii), we get @ F ` ’ g (we use the Fourier transform only in the variable Q ). Integrating this equation g initial a with @ ¿À , we get ½ ] @ 8 ¿½ #J condition À 9O La h g Taking the inverse Fourier transform, we  get [  o @ @ ½ ] Q 4`   L a h 8 g ¿½ ™ s c‘u  kœ L a h™‡ L W ¿À D J (1.13) [ h @  we have still to show the existence

@ ¿À 9Q pJ Of course, M ¿ Clearly, À 9O Q   ` of a solution. We skip the check that formula (1.13) gives a solution in  1° . This defines us a linear operator (the propagator)  ] k
Û 1—  Û 1° # ¿À iQ r ½ ] Q #J @ We would like to show that it is an integral operator and find its kernel. Let Ë= ] Q 4` `   L a h 8 g . Then  g [ €  - D @  [ û  [ o 8 g W  D ¿À  D @ Ëe ] QÍF ¿À s ‘ c u [ [ [   La h  L  ü   [  û [ o - D W 8 g D @ 8 g ¿À ) @ À ] Q #J À ¿ s ‘ c u `   L a h  L   La h [ [ L ü   8 not belong Unfortunately, this computation is wrong since the function   L a h g does to Û 1° . A way about it is to consider this function as a distribution and extend the D

(i)

B

B

D

D

D

E


F

E

;F

D

;F

#D

JI

KD

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8

NB

B

E

8

8

E

F

OE

GF

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PD

8

E

8

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H

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F

GF

H

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H

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Forier transform to distributions.



F

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E

LECTURE 1. QUANTUM FIELD THEORIES: AN INTRODUCTION

12

ï [ —1 \ ¿ 1

Recall that a distribution is a continuous linear functional on the space of smooth functions with compact support (test functions). Any function which can be integrated over any finite closed interval (but not necessary over the whole ) can be considered as a distribution. Its value at a test function is equal to

[ D

@

¿À M& À¿ 9Q &½ 9Q Q [  where the bar denotes the complex conjugation. Such a distribution is called a regular distribution or a tempered distribution. The rest are called singular distributions. We shall denote the value of a distribution ¿ on a test function & by [ D J @

¿À M& &½ iQ —¯¿ 9Q Q [  If ¿ is a regular distribution defined by a function ¿À iQ from 1— , then g Ö @

p J ¿À M& ¿ & v An example of a singular distribution is the delta-function  iQ°F whose value at a test v

1— function & is equal to &½ . It is also denoted by . A linear operator Ù
~(Š g [ — 1

= ( ( Ù p

with ï \ extends to the spacex of distributions by the formulaÖ Ù ¿À '& @ ¿À Ù & #J If ¿;R distribution, we have gÖ 1— , viewed as a regular Ù ¿À M& @ Ù & ¿ @ & )Ù ¿ so the two definitions agree. @ @ W be defined on the space of functions with@ square integrable For example let Ù Ù derivative. We have F Ù and for any distribution ¿ , ¿ w M& ¿À )F0& . If ¿ is a tempered distribution defined by an integrable differential function ¿ such that ¿ w also defines a tempered distribution, then the formula of integration by parts shows that this definition agrees with the usual definition of derivative. Since transform is an example of an operator defined on  1° with @ the, weFourier transform of a distribution ¿ by 4` can define the Fourier '¿ M& @ ¿À 4` M& ) pJ All the properties (i)-(v) extend to distributions. In property (v) we define the convolution of a regular distribution ¿ and an element & of Û 1° by the formula ¿ M& @ ¿À k& pJ v with Re v k{ O v @ O , Lemma 1.2. For any R  x W h @ s o c v   W h 8 ÷ x J 



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13

v Ij O

W to  1° and,  x h belongs Ð [ [   o o  x W h @ s c‚u  x W h   L W D Q @ s c‚u  4x W ‡  h   h 8 ÷ x D Q @ [ [ hö   os ss uv 8 ÷ @ s o 8 ÷ J c‚u   h x c v   h x [ 1— \ , Therefore, for any &;R ï [  o @  4x W h M& s c v   h 8 ÷ x &Å D J [ v@ v Consider the both sides as functions of . When Re Îj O each side is a holov v morphic function, and, for Re { O

O , are continuous functions. The unique continuation principle for holomorphic functions implies that the two sides are equal v v @ O . This proves the lemma. for Re ž{ O

Now we can use the lemma to set Ë= ] Q @ ` s o c‘u   L a h 8 g @ s o c‘u   L a    h 8 g @ s c‚o u ’  L ô h J ] h Property (v) of Fourier transform gives us [  @ @ ž ] ¿ Ëe ] Q k¿ Ëe ] QÍF - ¿À - D @ [  [ o ÿ  [ - - D J @ D

¿À Ë= Q ¿À [ s c‘u ’ ]  L  ô h " h [ ]   @ Ë= ] QIF - p ] j O

Thus we see that the integral operator with the kernel Ë= ] Q - is well-defined as an operator ž ] on the space  1° . Now observe that ž ] ¿ @ À ] Q @ `   L a h 8 g ¿À ) @ 8 g ¿Å @ ¿À @ ¿ J   La h This shows that ž ] is a unitary operator. In particular, ž ] is bounded on  1° (of norm 1) and hence continuous. It is known that  1À is dense in 1— . Thus we can Ög . Then the function Proof. Assume first that Re using the Gaussian integral, we obtain

B

Y

H

D

H

H

H

H

D

[Z

GF

F

'

'

H

D

H

#D





I

 \



















B

]

]

]

]

]

H

8

E

GF

]

]

H

]

D

E

GF

]

]

E

GF

]

]

B

B

14

ž ]

LECTURE 1. QUANTUM FIELD THEORIES: AN INTRODUCTION

extend by continuity to an unitary operator on the whole space the property

Ö g

1— . It satisfies

'ž ] ¿ @   L a h 8 g M¿ pJ Using property (iii) we get ž ] ¿ @  L a ¿ ] j O @ F W J This justifies the claim that Ëe ] QÅç ]Gw Q w is the kernel of the operator where ž ]Gw F ] @   L h ha É ša  on Ö g 1° . Finally let us try to justify the following formula from physics books: Ë= ] QÅç ] w Q w @ Q½ð   L  a É ba  ð Q w (1.14) First of all for any & from a Hilbert space , physicists employ the bra-ket notation &Àð k
@ & #J If Ù is a linear operator in , then &Àð Ù ð
@ & ™Ù pJ Let & be a normalized eigenfunction of an operator Ù with an eigenvalue . Physicists denote it by ð (although it is defined only up to a factor of absolute value one). To simplify the notation they set Åð6ð Ù ðð } @ ½ð Ù ð } #J H

D

D

@

.

U

U

@

@



/









/^

/^





ba

c



/_

`/_

c







c





c

Consider an operator (the position operator)


 1— Û g 1— # ¿  Qš¿ J Ö 1— It is a self-adjoint operator Û 1—  g . Its eigenfunctions do not belong to the space g 1— but rather to the space of Ö distributions. WeÖ have '& @ & @ 9Qš& @ v &½ v @ v '& pJ x x x x v Thus can be considered as an eigendistribution of with eigenvalue . Thus for any Q;R  1 É x we have, according to physicist’s notation, ð Q @ W . Now we have to compute . Recall that we can view it as an integral operator with kernel Ëe ] QÅç ]Gw Q w W   L a ša on the Schwartz space  1° which obviously contains ï [ 1— \ . We have defined  [ àp D @ àK #

Ë= QÅç Q  iQ F Q Ë= Q¥ç [ ] ]w w w w ] ]w  d

B

B

d

S

d

S

S

S

d

S



S

B

S

S

15

ð   L  a É ba  ð @ 7ð Ë= ] Q¥ç ] w àK
@ 9Ëe ] QÅç ] w v ) @ Ëe ] v ç ] w àp pJ v x @ Q /à @ Q w we get formula (1.14). Wex have to understand it as Taking W   L  a É ba  W É @ Ëe ] Q¥ç ] w Q w #J % For any function
  and a point ] R  one can consider a function on the  

 

set Map defined by % * ] ,ª '& @ % M&½ ] ™ #J % % Let KJKJJ4  be functions on  and ] JKJKJ¥ ]  R  , one can consider the integral ` % * JKJKJ¥ %  *  
@  ` % * JKJJ %  *  657 ( J , ] ,ª '&  L 3 & ], 64  ` ] ` ,ª '& ` ]` % % The right-hand side is called the path integral with insertion functions KJKJJ4  . The • left-hand-side is called the correlation -function. In the example above` W * ] , WBÉ * ] w , @ Ëe ] Q¥ç ] w Q w #J 

S

Sfe







Sfe

@

S

S

S

W





Map



S



S

Exercises

> 9ƒ @  [ W W ù D Q J [   )h ‡ L…„  Compute the coefficient at ƒ . g @ Ëe MO ] ç™Q  (defined to be zero for ] O ) 1.2 Show that the distribution Í ] Q - is a generalized solution of the equation F co WW @  ]  iQæF - #

a (you have to give the meaning of the right-hand-side). W is a generalized eigenfunction of the 1.3 Show that, for any { O , the function — 1

L operator ’ W in coincides with one of these Ö g and any generalized eigenfunction functions. v 1.4 Find the Fourier transform and the derivative of the Dirac function  9Q§F . 1.1 Find the Feynman rules to compute



g

U

8

U

c

b8





S

S



a

S

16

LECTURE 1. QUANTUM FIELD THEORIES: AN INTRODUCTION

Lecture 2

Partition function as the trace of an operator Ù

Ù

Ù

K  J K J 4 J

   `Ù @ Ž  Ù pJ (2.1) Tr  L L L ` unitary), then one can choose an orthonorIf Ù is a normal operator (e.g. Hermitian or % mal basis of consisting of eigenvectors of Ù . In this case @ Ž D

Tr Ù (2.2)  b D where Sp Ù is the spectrum of Ù (the set of eigenvalues) and is equal to the dimension of the eigensubspace corresponding to the eigenvalue . Notice that  J Ù @ ‰  b This gives @ Á ) pJ Tr Ù (2.3)  ‰ There are several approaches to generalize the notion of the trace to operators in infinitedimensional Hilbert spaces. We shall briefly discuss them. First assume that Ù is a bounded operator. First we try to generalize the definition of a trace by using (2.1). One chooses a basis JKJKJ4  KJKJJ and sets `  [ Ù p

Ž @  Ù

Tr  L L L 17` of an operator in a finite dimensional Hilbert space  is Recall that the trace Tr equal to the sum of the diagonal entries of a matrix of with respect to any basis. If we choose an orthonormal basis , then 



hc

c

a  Sp 

ic

j

c

l

k

U

c

a

a  Sp 

m

j



k





18

LECTURE 2. PARTITION FUNCTION AS THE TRACE OF AN OPERATOR

Ù @



if the series convergent. If the convergence is absolute, then this definition does not depend on the choice of a basis. In this case is called a trace-class operator. For example, one can show that Tr Gn(o Tr hopn if both n and o are trace-class. An example of a trace-class operator is a Hilbert-Schmidt operator in the space .  If is its kernel, then

Ëe 9Q 

 D} Ö g ç

 D} J @ Û Ù

Tr  eË 9Q Q

Ù

When is a self-adjoint Hilbert-Schmidt operator, the two definitions coincide. This follows from the Hilbert-Schmidt Theorem.

D} @ D} be@ a finite set o~ JKJKJ ™• equipped with the measure   1 with inner product Ö g   & @  & ¯  D~} @ Ž  v¯ à

L L - @ @ v v ` L K

K J  J 4 J

#

à 

K J K J 4 J /

à

 . It is clear that Ëe 9Q can be identified @  where & and ` with a matrix Ë ` L Ù &½ 9’ @ Ž  v

 L ` so that Ù is a linear operator defined by the matrix Ë . Then its trace is equal to  J Ž @  Ù

Tr   ` This agrees with definition (2.1) when we take the standard orthonormal basis of 1 . As we have already mentioned, in physics one deals with unbounded linear operaD~} like a differential operator. One tries to generalize definition (2.3). tors in g 

Notice Ö that Á ÙÛ @ Ž  Á @ F D  D  _ È  \ J  L L Z` L È _ ‰ ` Now for any Ù such that L has a basis of eigenvectors of Ù one can define the JKJJ  JKJJ be the sequence of zeta-function of Ù as follows. Let O Ù g positive eigenvalues of . One sets `[ MZ @ ~Ž  ›  

_ ` where ›  is the multiplicity of  , i.e. the dimension of the eigensubspace of eigen@ g  D~} , where  is a compact manifold vectors with eigenvalue  . When D Ù Ö ³

!



Example 2.1. Let n . Then



/^

Gn

q/

/

hr



s

s

r

s

r s4s

s

tm`j

m

k

s


u

hc

c





v



c



c



c



c

c

c

of dimension and



is a positive elliptic differential operator of order w , one can show

19

MZ j D ˆ O Ù @   É  \  J ‰ % This obviously agrees with (2.3) when is finite-dimensional. Also it is easy to see that for any positive number Ù @  \  ÙÛ #J (2.4) ‰ ‰ @ -Âà % . This of course agrees with the finite-dimensional case because 9O @ F W which acts on the space g ' , where Example 2.2. Consider the operator Ù @ ‚ c u D hh ° 1 ˆ Ö that` in this with the usual measure Q descended to the factor. Note measure ‚ m ‘ c u `D the length of  is equal to , i.e.  is the circle of radius . The measure ` Ù of metric  onWB8 the )• ` circle determined by its radius. The Q corresponds to the choice normalized eigenvectors of are . The positive part of the spectrum R m `  L ›  @Vc . Thus consists of numbers •Åˆ g )• R m g € \ with [ 'Z @dc ‘Ž  •Åˆ  g _ @tc g _ c Z p

` where MZ is the Riemann zeta function. It is known to be an analytic function for c §

j 7 o ˆ Re MZ . This agrees with the above since  is one-dimensional and Ù is an ` elliptic operator of second order. We have MO @ F co w MO @ F co Á c‘u J Thus w 9O @ F c Á c‘u #

MZ

v

is an analytic function for Re w and it can be analytically extended that  to an open subset containing . In this case we define j

x y

k

c

j

k

hc

x y

c

j

k

v

U

U

j



w

w

w

Q

w

v

z

C| {z

w



v

w

v

v

v

v

m

tm

w



w GF D D Q g @ c‚u g J (2.5) g @ 1°ˆ c‘u m and  @ 1°ˆ c‘u 7m . Example 2.3. Let us consider the path integral when  We use the action  g DJ o @  ž c \ € w ] g ] A map ;
Pá extends to a map of the universal coverings E
~1|á1 . It satisfies ] @ ] H c‘u rH c‚u • for /some integer • (equal to the degree of the map of

 be the set of maps corresponding to the same • . oriented manifolds). Let Map   /

 

 can be uniquely written in the form It is clear that each R Map • ] @ ] HÓ \ ] @  ] ÅHÓ \ ] #

and

det

w

3}



~

w



3}



w

}

w

20

LECTURE 2. PARTITION FUNCTION AS THE TRACE OF AN OPERATOR

where on such

\ ] satisfies \ ] H c‘u is equal to  g o @ 4 ž c \ €

3}

~

@ \ ] , hence belongs to g k . The value of  Ö \w ] g H c \w ] “ w ] ¥HÝ w ] g D ] J

 g \ ª D @ •  g \ D @ • \ c‚u \ ™ @ J \ € w ] w ] ] \ € w ] ] ì F 9O O Thus u  ž  @ • g g H co \ g € \w ] g D ] J The Minkowski partition function is > @  !ê‚ ( @ Ž h ªh  º !ê‚ ( ÀJ  €L   7L 3   7L 3 Observe that we must have  º ê7 (Í @  c‘u D

64  7L 3  Ëe MO Q¥ç Q Q where Ëe ] Q¥ç ] w Q w @ c‚u ’#o F  L W É  W h 8 g  a É ba 

]w ] We have

~

~

}

w

}

3}

w

~

w

}



Map

ƒ

‚

{€

Map

3}

Map

„

to be consistent with the previous computation of the path integral. This gives

os  D Q @ s Ž >d@ Ž J ’  €L h h   € L h h c‘u ’  Now we apply the Poisson summation formula Ž W  @ s o Ž  8 W J   š€ h Q   š€ h @ ˆ ’ , we get Taking Q  8 J (2.6) > g @ s Q Ž  W @ Ž  8™W @ Ž   € h   € h   € L h h @ Let us compute the trace of the operator g @  h h . Itswenormalized W78  . € ByL (2.1), eigenfunctions in g  are the functions    L € have Ö  ` L g 8 J

@ Ž  ð h  @€ Ž Tr g   € L h   š€ L h ™h L € 

ƒ

‚

{€

w

}



}

ƒ

‚

{€

{€

{€

}

w

;w

~

}

{€

{€

z

{€

~:… … 

~K@

/

~K@

 /

{€

~

Q

…

… 

/



z

{z

{€

~

z

21

>d@ Tr g p

L €

Comparing this with (2.6), we see that

~K@

@ F . ` h Remark 2.1. Ifg wea h repeat the computations for the Euclidean partition function (replacing  with ’G ) we get >? @ s Ž  8 @ Ž  8 J   € h ™h   š€ h ™h This shows that > ? @ s > ? ˆ pJ s , we get If we modify the partition function by inserting the factor o‚ˆ > ? @ > ? ˆ pJ where .

U

U

}

~

w

w

}

z

{€

}

~

w

w

z

{€

}

}

}

w

w

}

}

w

}

w

This is the first glimpse of the T-duality. † Let

9¾ @ Ž  € L  h » ¾ R @ Q H ’ R
æj O J @ Q ( equal to the value at be the modular form associated to the quadratic form iQ g zero of the Riemann theta function in one variable). It satisfies the functional equation GF o‚ˆ ¾ @ )Fž’ª¾ œ i¾ pJ h (the proof uses the Poisson summation formula). Observe that > ? @ 9’ ˆ g #J 

?5



W

!

{

{€

d

†

†

†

}

}

w

w

This is our first encounter with the theory of modular forms.

>d@ 

û F co  \ g € Q w ] g D ] ( 4  ‰p‹Œ ü is the circle of radius . Notice that  g D @ @ D

È Q w ] g ] QÅ ] Q w ] È \g € F \ € Q½ ] Q w²w ] ] F >?Ý@  º W~ ª W~ 

    a a º

There is another way to compute the partition function for the action

where   g \€ Thus

~

Map

Q½ ] p

~

}

~

~

ˆ

Map

@

G‰



QÅ ] p Q½ ] w!w   J h3œ  ‡

LECTURE 2. PARTITION FUNCTION AS THE TRACE OF AN OPERATOR

22

@ F . The integral g` a h h  ~W  ª šW~  ( #

  a a QÅ ] can be thought as a generalization of the Gaussian integral since QÅ ] # ™Ù QÅ ] is a @ @ % %

1 Ö ' ` . If  and Ù is a positive-definiteÙ self-adjoint quadratic form in g operator, we could use the orthogonal change of variables to diagonalize and write  º  D JKJJ D  @  º W W D Q JKJKJ D Q  @ Q Q    œ hœ   h ` ` Ð Ð   [ W D @  y u @ u  8 @ u  8 @ o Ù J   sg g Ù ‰ u žhœ  [  h QL  L ` L` L L` L ‰ Here we assumed that all eigenvalues are positive, or equivalently, that the quadratic @ ß ™Ù ß is positive definite. L To get rid of u let us change the measure on 1 form D D replacing Q with ` gº € Q so that  s u  D Q JJKJ D Q  @ Ù J  `  ‰  hœ ø Ù
 Now, for any normal positive definite operator in a Hilbert space , we v   can write any element &eR as a sum , where '&  is an orthonormal basis & v of eigenvectors of Ù . The coordinate  is an analog of the Q coordinate from above. This motivates the following definition   ( @ o 8 JL det w u (2.7) ï «  4` g Here the measure (+* , is defined up to some multiplicative constant ï . In fact we will be defining the correlation functions by the formula % ] # JKJJ %  ]  @£ò % ] K %  !ê‚  ] (+ *  L 3 ê7 (;* ,

` ` ò  L3 , ` ` so the choice of the constant will not matter. We would like to appy this to the op@ F in g 1Yˆ c‚u . However, not all of its eigenvalues are positive. erator h h formÖ the nullspace of this operator. If we decompose Constant functions each vector a v v as a sum    of normalized eigenvectors, then coefficients  will be analogs  of the coordinates in 1 . So, we can write our space as the product of the space of @ v \ constant functions and functions with O coefficient of the constant function o at \ @ o‚ˆ s c‚u is equals c‚u to s c‘u s .c‚u Thus@t. The the integral over the space of constant D ˆ c‚u s J functions is equal to ò \ g € >d@ c‘u Q c‘u s @ s So, using (2.5), we obtain 4` ˆ J where .

U

U

ˆ



G‰



ˆ‹Š



l

Š



a

‰

l

a





ŒŽŒŽŒ



c

d

a 





„

c

j

k

j

k

c



Q

ˆ‹Š





Š

j

‰



k



u



ˆ‘

’

‘

d

‰ “9

9



.

/

u

U



3}

U

/

3}

3}

3}

w

{z

3}

w

}

}

w

}



23 This agrees with the computations in example 2.3 if we switch from the Minkowski partition function to the Euclidean one.

 

] Ö @ ‘
ÙÚ

]  Y H 1 ] ]   @ ž ¥H co g  ÈÈÈ ê  ê  g H terms of higher order in ½J g This gives a semi-classical approximation:  i’Gž 4 ˆ . )(+* , Ž i’Gž ˆ . p* c‚uo ’ g  ÈÈÈ ê  ê , œ J ‰p‹Œ ‰p‹Œ ‰ g h This approximation is exact when the action is quadratic in . @ 1°ˆ c‘u and  @ 1 . The Lagrangian is Example 2.4. We take  ã Ö ã Ø @ co ã Ø g F g ã g Ö

Here is another application of the Gaussian integral for quadratic functionals. Consider the action functional defined by some Lagrangian . We know ” that its stationary points are classical solutions. Write , where S ” is a classical solution. Then S

”

S

q•

GS

S

j

”

—–

k

S

S

classical solutions

•

™˜

>  @  +( * ] , û c’  Ø ] g F g g ] D ] J p‰ ‹Œ  ü It is called the path integral of the harmonic oscillator. The kernel of the operator   is given by   L aªša  ê   W (;* û ’  Ø @



\ Ë= ] ç ] Q ê a  œ  ] , c  ] g F g g ] D ] J ‰p‹Œ ` ü a and the Minkowski partition function is

š˜

A @



A

š˜

3R

Choose a critical path cl for the action defined by our Lagrangian and decompose the action in the Taylor expansion at cl .

ž ] ) @ ž ¥ H co g  ÈÈÈ ê  ê ] F #J g The classical path is a solution of the Lagrangian equation: D â â ã â âã @ H @ J D ] ÖØ F Ö ] w g ] O @ š ) ] @ Q is Its solution satisfying the initial condition ] \   ` ] @ û  00 ] ] ` F F ] ] \ H Q û  00 ] ] F F ] ] \ \ J ` ü ` ü S

cl

cl

S

cl

˜



cl

 tm



 m

˜

˜

 tm

 tm

˜

˜

(2.8)

LECTURE 2. PARTITION FUNCTION AS THE TRACE OF AN OPERATOR

24

û g H Q g 0 ] F ] \ F c Q  o @ @ D J ž c a œ Æ ] w g F g ] g ] c  0 ] F` ] \ ü Ç ` a of the action functional The second variation is g  ] ™ @  a œ ] D D g H g ª ] D ] J ]g g a Thus we can rewrite (2.8) in the form Ëe ] \ ç ] Q @ i’Gž ) Ü  ê   W (+* û ’  ` ‰p‹Œ ) D g H ) D J ê a  œ  ] , ‰K‹<Œ F c a œ ] F ] D ] g g ] F ] ] ü a @ a Now let us make the variable change replacing ] F ] with ] . The limits in the @

\ path integral change to ] we integrate over are periodic ] \O . @The paths @ ] F ] \ satisfying @

with the period Ù ` ] we have] O . Using the generalization of the Gaussian integral to functional integrals  ê   ` \ (+* û ’  ` D g H “ D @ ê a  œ  \ , ‰p‹Œ F c a œ ] D ] g g ] ] ü a  a º (+* o À ’ ( N @ c‘uo ’ (æ

c , ) F  hœ   ‰p‹Œ ‰ h where ( @ F DD g F g J ]g \ @ @ ( The eigenfunctions of satisfying the condition the functions ] corresponding ] O areeigenvalues  • u ] ˆ7Ù , where • Rtm \ and Ù @ ] F ] \ . The are ` @ • u ˆ‚Ù g F g . We know that` equal to  [ [ [ c‘uo ’ ( @ ‘ c‚uo ’ ) u •Åˆ7Ù g F g @ ~ c‘uo ’ • u ˆ7Ù g ‘ o F • g Ùu g #J ‰ g g ` ` ` Of course here we use a “physicists’s argument” since we don’t have the right to write the product as the product of two infinite products , one of which is divergent (see the next remark for an attempt to justify @ the argument). Now we use that the first O . So to be consistent with our previous product corresponds to the action with computation we must have û [ c‘uo • u ˆ7Ù g  œ @ Ëe ] \ O-ç ] O @ s c‚ou Ù J ’ ‘ ’ h ` ü ` The value of the action functional on the classical solution is

cl

™˜

cl

A

 m

cl

S



3R



˜

˜

˜

A

S

A

›[œ 



˜

cl

˜

cl

A

cl

cl

A

˜

A



j

3‡

k

Map

™˜

 tm

&|

j

c

™˜

l

k

š˜

l

˜

l

l

˜

25

&½ ] [ ~

F ` Wh g€L h

*] \ ] , &½ ] \ @ ½& ] @ O ` ` c‚uo ’ • u 7ˆ Ù g @ c‘u ’ Ù ï J

Note that if we compute the product using the zeta function of the operator on the sapce of functions on satisfying we get l

ï @ ï s c‚u

`

U

U

The two computations disagree. The way out of this contradiction is the choice of the normalizing constant which we used to define the Gaussian integral. It shows that we have to choose . Now we use the Euler infinite product expansion for the sine function:

 Ù @ [ o Ù pJ Ù ‘ F • gg u g g ` From this we deduce that Ëe ] \ ç ] Q @ i’Gž ™ c‚uo ’ (æ ` 8 g @ ‰ ` ‰p‹Œ i’Gž ) c‚u ’  ٠` 8 g @ Æ u œ s o  F L -8 g 9’Gž ) pJ ‰p‹Œ Ç h   g L ‰p‹Œ Let us rewrite ž in the following form ž ] ™ @ ’ c û iQ g H g oo HF   g L F o QF   L @  gL   gL ü ’ c Æ iQ g H g o H c Ž [ g  F Q o H Ž [ g  @ ‘   L   L ‘   L Ç `[ ` Ž  J ’ c û F| iQ g H g ÅH c Q g H c g F Q

g ‘ \ L  L ü  tm

l

;˜

˜

˜

j



k

cl

˜  tm

cl

Ÿž

˜

cl

ž

˜

cl

i˜

i˜

¤



ž

 

cl

ž



ž

ž£

¢¡

ž

ž

ž

¡

h˜

¥





ž

ž£

¡

Using this we obtain

Ë= ] \ ç ] Q @ Æ u œ s o  F L -8 g Ü Ç h  gL ` [ c H c Ž û H 

H F c F| iQ g g ‘ \ Q g g F Q   L   g L  @ ‰K‹<Œ ü Ž [ š  i  iQ - @ N8 ˆ u šW 98 HdJKJJ4J ‘   L ‡ hœ   L g   h)‡ h g ` ¦ž

˜



§

ž

˜

¥





ž

ž

¡

ž

G

n



ž

„

˜

ž

R

¨

26

LECTURE 2. PARTITION FUNCTION AS THE TRACE OF AN OPERATOR

Now recall that the kernel of a Hilbert-Schmidt unitary operator can be written in the form

Í iQ - @ Ž  ¯  iQ   p

where  is the normalized eigenfunction with the eigenvalue  . In our case, the ¥   . Thus the eigenvalues of the eigenvalues of must be equal to Î • H

 L u 0 ÷ ` s . We shall  see L in‡Ûthehœ lecture that the eigenvectors of Hamiltonian  are @

 8 are@ the\ Hermite iQ ˆ `\ 8  g \ -° Q@   W h ˆ -u 8 g where are polynomials. @ •

#

J š   W 9  This checks the When O , we get 9Q i Q   hG‡ h g first term.



g

/

c

4/



{€

/

@



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/

.

c

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n

Exercises

® ±
O @  g  JJKJ MZ `  _ MZ Z O @ O [ @ É \  J `  for any ­ {to , [  @  KK [  . L  that L (i)` Prove ë L ë ‡Å` L g s ` L @ ‘ c u ` L – (ii) Give a meaning to the equality . @ 1  ˆ X be a • -torus. Here X @ m HVJKJJ m  and JKJJ  are linear 2.2 Let Ù  independent vectors in 1 . ` @  â Ð` D} (i) Compute the trace of the Laplace operator D} is induced by the standard volume formÖ on 1 F  . L  ` Wg in Ö g Ù0 where > ? Ù for the maps from  @ 1°ˆ c‘u (ii) Compute the Euclidian partition function @ ò  ðð w ] ðð g D ] , where is any lift of to Ù with the action defined by ž   | 1 U  1 ` to a smooth map . g @  1
 QARÓm for all QARÝX . Use the Poisson summation (iii) Let X R formula Ž ¿À iQ @ Ž ¿½  p

4` ‘¬ W ‘¬ @ * KJJKJ  , , to relate  > where ¿+R  1 , ¿ is its Fourier transform and > ? ? the partition functions Ù and Ù K . ‰ ` 2.3 Compute the terms 9Q - and g 9Q  from Example 2.4 to find the eigenfunc

tions 9Q and 9Q . ` g `



be a non-decreasing sequence of positive real numbers. 2.1 Let c c v  6 Define provided that this sum converges for Re `ª and has u c a meromorphicx continuation to the whole complex plane with no pole at . Set «

c



c

ic

c



c

c



+˜

^˜

˜

˜

šu

3}

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!

Ÿ

E

n

R  %®



E

:B

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/



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n



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j

k

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Lecture 3

Quantum mechanics O Hdo

The quantum mechanics is a dimension QFT. Let us recall the main postulates of quantum mechanics. A quantum state of a system is a line in a separable Hilbert space  . It can be represented (not uniquely) by a vector / of norm 1. To each observable quantity (like position, momentum or energy) one associates a self-adjoint operator n in  (an observable). A measurement of an observable n depends on the given state / and is not given precisely but instead there is a probability that the value belongs to a subset c . This probability is equal  4/ ° hc to ¯—° ic / , where ° ic is the spectral function of n , an operatorvalued measure on . In the case when n is a compact operator,  has an orthonormal ²± basis of eigenvectors of n with eigenvalue c . Then ° ic u  , a  a ²± W where  is the orthogonal projector operator to the subspace . Thus, for any   simple eigenvalue c of n , / can be interpreted as the probablility that the observable n takes value c in the state / . / In physics literature one often writes c  for a norm 1 eigenvector a of n with  /  /     eigenvalue c and rewrites ba in the form c . Also one writes c instead of   . c The probability amplitude (a complex number of absolute value ) is defined to  /  /  is called  the wave function of the state / with be c . The function c c respect to n . The inner product of two states /^ is interpreted as the probability amplitude that the state changes to the state / . Its absolute value is the probablity of this event. Note that by Cauchy-Schwarz inequality, this number is always less or equal to 1 and it is equal to 1 if and only if the two states are equal (as lines in the Hilbert space). The expectation value of n in the state / is defined to be

GF -,   @  

  k ç @ 6ð ð   k1 6ð ð g      ð 0   ð g ð 0

& ½ðð6ð } 0

 

&

n  

Åð } o

ð

&

@  @ =[ Q D}  —

(3.1) [ @   D~}  is the measure on 1 defined by D}   9¦ M¦  . where @ g 1r D Q and @ is the position operator corresponding Example 3.1. Let Ö coordinate Q . It is defined by M¿ @ Qš¿ . @ This is an unto the measurement of the bounded self-adjoint linear operator. We know from Lecture 1 that ð (although °



n

³



n

/

/_

°

³



d

n

°

³



³

°

4/

/_

d

c

27



Sfa

LECTURE 3. QUANTUM MECHANICS

28

iQ # @  i Q D Q @ pJ The probability amplitude of the value of the observable in a state is equal to . So, in the realization of as Ö g 1 D Q , a state iQ is interpreted as the wave function of the state with respect to the observable . Any R can be written as @ D J they do not belong to the space 

but rather to the space of distributions). We have

 /



S´a

/

/

S´a

d

c

/

ic

d

/

&

/

/



/

ic

/

hc

S a

/

µ

c

Of course this has to be understood as the equality of distributions. For any test function we have

[ [

û  [ D ½& iQ D Q @ [ ü   [ &½ [  The expectation value of is equal to @ º /

hc

S´a

[ û [ D D @ &½ iQ Q [ [ ü  

D @ M& #J /

c

/

hc

hc

ic

Sfa

c

/

c

d

 d

½ð ð g D J

 ³

/

c

hc

c

1 D Q ™ g

Ö x x W  8 x g ` €‘a   x h g a ] x

k
@ Á6Âà \ s c‘o u W  8 g @ Á6Â6à \ s c‚o u   8 g J ]    x h a a Ä ]   x h a x a Ä @ à When Q is a variable, we get

W @ Á6Â6à \ s c‚o u W  8 g @ @  9Q§F v pJ x @ 1°a Ä ˆ c‚u ]  4x h a x  @ ’ W . Example 3.2. Let and equal to the momentum operator g @  Ö Its eigenvectors are the functions ¿  `   L WB8 with eigenvalue • . We have g€  o iQ p ¿  @ s c‚u \ g € ¯ 9Q   L  WB8 D Q @ v¯ 

à

Consider the delta-function S as a state (although it does not belong to .  d Then the probability of to take a value in the state S is equal to S Sfe  . The inner product is of course not defined but we can give it the following meaning. We know when tends to that S is equal to the limit of tempered distributions Q zero. Thus we can set 

S



S e



e

S e

S

S



S

3}



n

Q

 /

3}

U

~

~

~



S

/

~

U

29

@m •  

•+ v v¯  ð  ðg .

¯   •   @ Ž • ðv  ðg J 

is the -th Fourier coefficient of / . So, the wave function of / is the function on Sp . The probability that takes value at the state / is equal to The expectation value is equal to 

 ³

{€

Ð

The dynamics of a quantum system is defined by a choice of a self-adjoint operator . , called the Hamiltonian operator. In Schr¨odinger’s picture the operators do not change with time, but the states evolve according to the law

@ $ 0J a  a Here . is a fixed constant, the Planck constant. Equivalently, ] is a solution of the Schr¨odinger equation: ’G. D D ] @  ] #J a In Heisenberg’s picture, the states do notÐ change with Ð time but the observables evolve according to the law I ] @   $ a    $ a J We have the Hamiltonian equation: D (3.2) D ] ] @ * ] # S, :

where * l, : @ . ’ Ð F #J @ $ and hence the corresponding state If is an eigenvector of , then a with time, i.e. iQ describes a (equal a to the line spanned by ) doesa not  change @

/

/

/

/

@

n

n

n

/

.

f/

.

@

n

n

o

Gn

<

.

o

a

o

/

<

n

/

/

/

stationary state. Usually one measures observables at the stationary states of . .

@  D~} Ög

There are two ways to define a quantum mechanical system. One (due to Feynman) uses the path integral approach. Here we take  as in Lecture 1 and define the Hamiltonian by means of the path integral. The choice here is the action functional. It is defined in such a way that its stationary paths describe the motions of a classical mechanical system. Another approach is via quantization of a classical mechanical system. Recall that the latter is defined by a Lagrangian D which, in its turn, defines an action functional on the space Map

ž  @



x

e

p šØ ) D J Ö ] ] ]

@


ÅÙÛ  1 * Öv /à ,  #J

A critical point of this functional defines a motion of the mechanical system. The equations for a critical point are called the Euler-Lagrange equations. If one chooses

LECTURE 3. QUANTUM MECHANICS

30

ã ã ã the coordinates ã Åã Ø @ ã K JKJJ4 㠏 ç @ ã Ø K JJK` J4 K ¥JKã JØ  J4 in Ùin  (so and ã corresponding

@ , the local that Ø equations look as ` ` â D â @ @ L o KJKJJ4 )•—J L âšãÖ F D ] â ãÖ Ø O ’ (3.3) Ð L atL a path ]  ] p Ø ] ™ in ÙÛ given by ã @ Here the left-hand side is evaluated ã ] p ã Ø @   . For example, if we assume that the restriction of to each tangent L  Ù  a Ö space is a positive-definite quadratic form, we can use to define a Riemannian L L W a Ö metric on  . A critical path becomes a geodesic. Another way to define the classical mechanics is via a Hamiltonian function which is a function on the cotangent bundle Ù K . % defines a Recall that any non-degenerate quadratic form on a vector space % space a quadratic form as a symmetric quadratic form % . If we% view ` on% theas dual bilinear form, and4hence a linear map  , then is the inverse map. Let us 4`
% isUequal see that, for any ÝR , to the maximum`(if j O ) or the minimum 1 defined by O ) of the function Ì (if Ì  @  F  #J % @ % @ 1  and  @  , then  @ If we choose the coordinates so that  @  for some . Thus to find an extremum we must have äg` Ì +F  @ symmetric O , hence @ matrix4` and we get o Ã Ì ¥  @  +F c  @ co  @ pJ «‹ ` 4` 4` 4` % Using this one can generalize the construction of for any function ¿ on such ` the Legendre transform of that its second differential is non-degenerate. This is 4called ¿ . By definition, for any AR % , @  FÓ¿À < p

Leg '¿ @ D ¿ , where D ¿
% ^1 is where is the implicit function of defined by % . In order that this function be defined the differential of ¿ at the point R D g ¿À  ) @ O .weIn have to satisfy the conditions of the Implicit Function Theorem: general, the implicit function is a multivalued function, so the Legendre transform ‰ is defined only locally in a neighborhood of an extremum point of the function  F

. ¿À We shall apply the Legendre transform to the Lagrangian function . We denote Ö the local coordinates in the cotangent bundle Ù p by ã @ ã KJKJJ4 㠏 ç KJJKJ  #

KJJKJ4`  are Ùtaken ` to be the dual of the coordinate where the fibre coordinates ã ã Ø K

K J  J  J 

Ø

  in the tangent functions and can be identified with a basis ` bundle KJKJJ4 ` in Ù W . The Legendre transform of is equal to Ö  Ž œ A ã @  ã Ø F Ö ã ã Ø p

L` LL local coordinates

S s

L¶ ¶¸·i¹

U

·U





d

d



d





º

d

d

d

Gº

²»



9

»

d

9

º

9



9

n

%9

²»

º

n

“¼ ‘ 

q9

²»

º

d

f9

º

9

n

9

9

9

º

n

n

º

º

=º

n

º

3º

n

d

º

hº

d



º

hº

9

9

9

º

9

º

‘

º

j

‘

9

9

p'

Gº

º

9



¯

G¯

½¯

“¶ ¶¸·

k

¯

¶  ¶¸·

.

¯

¯

¯

9

where

ãØ L

K JJKJ4  defined by the equation @ ââ ãÖ Ø ã Å`ã Ø # ’ @ o~ JKJJ4 )•
Ù ÚU1 L is calledL the Hamiltonian associated to the Lagrangian

31

are the implicit functions of

½¯

¯

¯

Ö

 The function . . As we have explained before, in order it is defined the Lagrangian must satisfy some conditions. Using the Hamiltonian one can rewrite the Euler-Lagrange equation for a critical path of the action defined by the Lagrangian in the form:

]

Ø
@ D D @ F â A⚠ã ã ã Ø
@ D D ã @ â Aâ ã J ] L UÙ K L  L # ] L L Î
Ò ] which Here a solution is a path satisfiesL the above equations ] ã



<
Ù K Õ1 . The projection after we compose it with the coordinate functions  of the path to the base (i.e. the composition with the projection map Ù KÚá )

is the path ] describing the equation of the motion. The difference between the Euler-Lagrange equations and Hamilton’s equations is the following. The first equation is a second order ordinary differential equation on ÙÛ and the second one is a first order ODE on Ù p which has a nice interpretation in terms of vector fields. Recall that a (smooth) vector field on a smooth manifold is a (smooth) section J Let ï [ Ù denotes the set of vector fields. of its tangent bundle Ù It has an [

obvious structure of a vector space. For each smooth function &dR|ï one can [ Ù by the formula differentiate & along R ï ( M& iQ @ Ž â â Q &

L L

K

K J  J 4 J

L 9Q Q  are local coordinates in a neighborhood where QdR of Q , and are K

 J K J 4 J

of Ù W .L We the coordinates of` - iQ R Ù with respect to the basis W W W also have œ ( M& @ D &½ p

¯

¯

.

¯

.

¯

¯

&







¯





&

¾

F

¾

¾

;¾

F

¾

H

F

¿¾

F

F

¾

¶

H

¶

¾

¶

¶ 

;F

where we consider smooth 1-forms as linear functions on vector fields. This defines a linear map

(ý
ï [ Ù — End Mï [ ™ #J @ ( M& ×H§( k & , so that the image of ( lies in the It is easy to check that ( M& [ . Given a smooth map e
Å* v à ,  , subspace of derivations of the algebra ï and a vector field we say that satisfies the differential equation defined by (or is an integral curve of ) if D
@ D  Å â â @ - ™ v for all eRÎ /àp #J D] ] The vector field on the right-hand-side of Hamilton’s equations has a nice interpre@ Ù  J tation in terms of the canonical symplectic structure on the manifold ¾

H

/

H

;¾

/

H

/

G¾

¾

F

F

F

»

F

hº

º

¾



LECTURE 3. QUANTUM MECHANICS

32

Recall that a symplectic form on a smooth manifold is a smooth closed 2-form PR g which is a non-degenerate bilinear form on each Ù W . If we view W as Ù  Ù W @ Ù W , then its inverse defines a linear isomorphism a linear map W


-Ù  Ù W . Varying Q we get an isomorphism of vector bundles ÙÛ  Ù p i Q and byW the pull-back of sections an isomorphism of the space of sections
@ ï [ Ù ° ï [ Ù #J `  1 , its differential D Ì is a 1-form on , i.e., a Given a smooth function Ì
section of the cotangent bundle Ù . Thus, applying we can define a section D Ì of the tangent bundle, i.e., a vector field. It is called the Hamiltonian vector @ Ù  

with Ì field defined by the function . We apply this to the situation when ã ã



coordinates and Ì is the Hamiltonian function A . We use the symplectic form given in local coordinates by @ Ž Dã D J L L b

Ù

L For any R W W š  @ Ž D ã < D  F Ž D ã  D < pJ L L L L L L In particular, â â @ â â @ W âã âã W â â O for all ’ N

L â â L â â @ W âã Ð â F W â Ð âã @ J D @ L D ã @ F L hence L This shows that â L â â â â Lâ D @ û Ž âã D ã H Ž â D @ Ž GF âã â H Ž â âã J L L L Lto theL ü Hamiltonian L L L vectorL field L  L defined L corresponding So we see that the ODE by is the vector from the right-hand-side of Hamilton’s equations. We have  M¿ @ ¿ J [ one Let be a symplectic manifold. For any two functions ¿ R|ï defines the Poisson bracket ¿ @ 0 D ¿ p D - ) pJ D @ D ¿À , so that By definition of we have 0 ¿ p ¿ @ D ¿À D - ) @ D  M¿ @ F D ¿  pJ ¾

˜

ÁÀ

 ž 

G¾



¾



¾

¾



˜

¾

Â

À

ž



G¾

¾

Â

;¾

¾



¾

¾

¾

Â

;¾

ž

ž

&

¾

¯

.

˜

¯

¯

ŸÃ

9 Ä

G¾

9 Ä

˜

9

˜

˜

s

˜

Â

Â

ž

Â

.

VÂ

¶ ¶¸·

9

¯

iÅ

¯ s

^˜

S s

s

¯

ƶ ¶[Ç

ž

.

.

ž

Ä

¯

¯ s

¯

ž

Ä

¯

.

¯

¯

.

¯

¯

È

G.

.

È

;¾

É

G.

!

.



˜

 !

Â



˜

ž



 !

Â

˜

Â

ž

F

ž

Â

Â

ž



ž



Â

;F

ž



Â

ž



G¾

33

The Poisson bracket defines a structure of Lie algebra on tional property :

ï [ satisfying the addiG¾

¿ @ ¿ ×H ¿ šJ One can ã showã (Darboux’s Theorem) that it is always possible to choose local coordinates KJKJJ  KJJKJ4  such that ` ` @ Ž Dã D J L L L In these coordinates â â â â ¿ @ Ž âšã ¿ â F â ¿ âã pJ L L L L L For example,

@ ã ã @ O ã @ J (3.4) L L L L The corresponding differential equation (= dynamical system, flow) is D¿ @ ¿ J D] v A solution of this equation is a path ;
N* à , á such that D ¿À ] ) @ ) pJ ¿ ] D] This is called the Hamiltonian dynamical system on ã (with respect to the Hamiltonian @ Ù  , we obtain function ). If we take ¿ to be coordinate functions on v  . Hamilton’s equations for the critical path ;
N* à , ´ L in L 

The flow of the vector field ¿ a by the formula ¿ is a one-parameter group of operators on æ  defined a M¿ @ ¿ iQ @ ¿À ] ) a a ¿ with the where ] is the integral curve of the Hamiltonian vector field ¿  @

Q , The equation for the Hamiltonian dynamical system defined initial condition MO by is D¿ @ ¿ J (3.5) D ]a a Here we use the Poisson bracket defined by the symplectic form of  . A quantization of a mechanical system is defined by assigning to any observable operator This operator may contain a parameter ¿A. . RÝThisï [ assignment a self-adjoint R .properties. must satisfy some natural For example: ¼@ Á6Â6Ã (3.6)  : \ * ,: ¼ Ä ¼ Ê !



¯

 ! Ê



ËÊ ! 



¯

˜

¯

¦Ã



 !





¯



¯

¯ s

!

>

s

¯

!





S s

!

.



!

¯ s

!

.

¾

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5

¾



!

.

Ì

5



.



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n

n&Í

Î[Ï

.

!



n

n

Î

.

!

LECTURE 3. QUANTUM MECHANICS

34

Under the quantization the Hamiltonian function of the mechanical system becomes a self-adjoint operator . , called the Hamiltonian operator of the quantized system. We have

  @  @ : Á6Â6à \ * , : @ : ÁÂ6à \ * , : J ¼ as the quantized ¼ action of the Hamil¼ ¼ Ä Ä Thus the linear map T* , : is interpreted tonian vector field  . The analog of the dynamical system (3.5) is the Hamiltonian equation in quantum mechanics (3.2).  For example, when a mechanical system is given on the configuration space Ù 1 ã with coordinate functions

we need to assign some operators to the coordinate functions: L@ L Ð   @ Ð @

’ o KJJKJ4 )•—J By analogy with (3.4), weL should haveL *    , : @ * , : @ O *   , : @ (3.7)   L @  Hilbert L space % andL operatorsL R E % satisSo we have to find an appropriate % 1

fying (3.7). We take L L Ö g
 andã define â & &  
&  ’“. âã & J L self-adjoint L L operators. L Recall that these are unbounded   (resp. ) is called the position (resp. momentum) operator . The operator L ofL quantization of a classical mechanical system given by a Let us give an example harmonic oscillator. It is given by the Lagrangian Ö ã ã Ø @ c—o i› ã Ø g FΛ g ã g ) p

where › is the mass and is the frequency. The corresponding Hamiltonian function is A ã @ ã Ø F co i› ã Ø g FE› g ã g @ co › g H › g ã g #

@ @ › ã Ø to express ã Ø via . The function e ã can be where we used that @

nÑÐ

@

nÍ

n

È

n

n

Ï

n

@

n

.

.

h.



¯

d

n

n

·

d

s

d

Ç

d

s

s

S s

d



d

d

µ˜

˜

.

¯

¯

¯

:˜

:˜

‡

¯

¶

¯

viewed as the total energy ¶= of· Ò the system. The corresponding Newton equation is

.

½¯

DgQ @ F Q J D ]g g So the motion does not depend on the mass but the total energy does. In the sequel we @ o . The Hamiltonian operator can be written in shall assume for simplicity that › the form @ co   g H c g g @ v
(3.8) ^˜

.

.

˜

d

Ó

˜

Ó

Ԙ

35

v @ s o c H ’   # v @ s o c FΒ  

where

d

;˜

Ó

d

©˜

(3.9)

are the annihilation and the creation operators. We shall see shortly the reason for these names. They are obviously adjoint to each other. Using the commutator relation d , we obtain

*   , @ Fž’G.

* v v , @ * v v , @ O q* v v , @ .

* v , @ F0. v q* v , @ . v J v v form a Lie algebra This shows that the operators o~

Ó

Ó

.

Ó

Ԙ

.

Ó

Ԙ

(3.10a)

Ԙ Ó

(3.10b)

Ó

 , called the extended Heisenberg algebra. . So we are interested in the representation of the Lie algebra  in Ó Suppose we have an eigenvector / of . with eigenvalue c and norm 1. Since is adjoint to , we have .

v ½ðð ð6ð g @ 0

c

 /

/

v

@  v v ¥H 0 . c @ ðð v 6ð ð g H . c 6ð ð ðð g J  /

/_ .

gÖ 1—

Ó

 /

/^

Ԙ

/_

Ԙ

/

/

This implies that all eigenvalues c are real and satisfy the inequality

{ .c J (3.11) v @ O . Clearly any vector annihilated by v is an The equality holds if and only if eigenvector of with minimal possible absolute value of its eigenvalue. A vector of norm one with such a property is called a vacuum vector. v @ F0. v , we have Denote a vacuum vector by ð O . Because of the relation * , v @ v FÓ. v @ ÍFÝ. v 0J v is a new eigenvector with@ eigenvalue FP. . Since eigenvalues This shows that v v v @ O O are bounded from below, we get that for some •P{ O . Thus  v ‡Å` that the existence of one eigenvalue of is and is a vacuum vector. Thus we see equivalent to the existence of a vacuum vector. v  Now if we start applying to the vacuum vector ð O , we get, as above, eigenÝ H • . . So we are getting a countable set of eigenvectors vectors with eigenvalue : g  @ v  ðO @ g  . J It is easy to see, using induction on • that that with eigenvalues  @  we obtain a countable set of orthonormal ðeigenvectors 6ð  ð6ð g •—– '. J After renormalization g ‡Å` ð • @ M. o  •—– v  ð O # • @ O o~ c KJJKJJ (3.12) Ԙ

c

/

.



.

/

.

.

/

/

Ԙ

ic

/

/

Ԙ

c

/

Ԙ

/

/

.

Ó



Ԙ

Ó

/

/

Ԙ

c

Ԙ



Ԙ

=

„

Ó

Ԙ



LECTURE 3. QUANTUM MECHANICS

36

Ö g

1—

ð•

One can show that the closure of the subspace of spanned by the vectors = is an irreducible representation of the Lie algebra  . The existence of a vacuum vector is proved by a direct computation. We solve the differential equation

s c v @ FE’   @ ã EH . D D ã @ O /

©˜

d

/

/

/

˜

and get

ð O @ . ùœ   h 8/: J (3.13) In fact, we can find all the eigenvectors ð • @ Æ . u ` 8 ÷  s 㠈 s . p

Ç where  9Q @ s c o  •—– Æ QÍF D D Q    ô h Ç h • is a Hermite polynomial of degree . It is known also that the orthonormal system of ÿ is complete, i.e., forms an orthonormal basis in the Hilbert space functions  iQ ô h • an irreducible representation of with unique vacuum we constructed h g 1— . ð O Thus p J Övector •§H ` . .The vectors ð are all orthonormal eigenvectors of with eigenvalues g function (3.13) gives the probability amplitude that a particle occupies the The position Q on the real line in the vacuum state of the system. According to Example 3.1, the value of the function ð • at is equal to the probability that the observable takes value at the state ð • . Finally let us compute the partition function of the Hamiltonian . The eigenvalues @ •ÍH . and their multiplicities are equal to 1. So of are  g` [   M: @ ã

Ž @

Tr (3.14)  L a ‘ \  L a ‡ hœ o F hœ ã ã@ : J where  La ˜



=

˜

ž

.

·

˜

.

.





=

.

Ԙ

=

d

=

c

c

.

.

c

Ԙ

@

՞

՞

Exercises

 

ð • ) Ù 1— ã ã ã ã @ Å

Ø

Ø 3.2 ã Consider the Lagrangian Ö i` › g F on , where ã @ O ã @ v



o for R MO and otherwise. g Quantize this mechanical system, solve the Schr¨odinger equation and find the stationary states of the Hamiltonian operator.

3.1 Consider the quantum mechanical system defined by the harmonic oscillator. Find the wave function of the moment operator at a state = . 5

5

5

37

@ W. À ¿ i Q @ F id g be the expectation value of the operator F 3.4 Let id (the dispersion of an observable at the state ). Prove the Heisenberg’s g Uncertainty Principle Î{ c°o ð * ×, ð J 3.3 Compute the Legendre transform of the function 

n

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Ö

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³



Gn



n

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 ³

/

Ö

n

Gn

³

Ö

Go

³



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o

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38

LECTURE 3. QUANTUM MECHANICS

Lecture 4

The Dirichlet action Â6à zjzo  @ Ù Ü­ Ù P1  ­ ` &
<| @ o~ KJJKJ )•

p

0 Ù

&½ ] Q ] R QÓ RÝ­ ­ O ­ &½ ] Q   M & ` ] p KJKJJ &  ] ) ž
ÙÔ´  1 o~ K JJKJ1 )• Ü o KJKJJ )• • ­ 
ÙTá1  Ù M& D & k
~ÙeÙ ¨ ÌÍÖ M& 

D} ÙÛ D} Ö   k 

 ÌÍ ® Â6à dj o

ÌÍ M& ÙÛ ®  ÙÛk j

, for example, Now we shall move to QFT of dimension larger than 1, i.e. where , or and is a manifold of positive dimension which we shall assume for simplicity to be orientable. A map is given by a function ! × . Note that when is -dimensional, say , we can view as a vector function and get the quantum mechanics on ! . For example, if our QFT is a harmonic oscillator, passing from  corresponds to considering harmonic oscillators. Replacing !  by positive-dimensional means that we consider the whole manifold of harmonic oscillators! Recall that any QFT is defined by an action functional on the space of paths. In a one-dimensional theory we defined by a Lagrangian . The pull-back of under the map is a function on , so for any density on (i.e. a section of j top  ) we can multiply to get a density on which we can integrate. If this is not true anymore since is a function on and a density is a function on 7ØqÙ Ú . So the definition of the Lagrangian has to be changed. We are not going into a rigorous mathematical discussion of this definition referring to Deligne-Freed’s lectures at the IAS. Recall that the jet bundle of order r of a fiber bundle over a manifold ¾ is a vector bundle whose local sections are local sections of together with their partial derivatives up to order r . Let be a local frame of and local coordinates on ¾ . A local frame of z is a set where  ´Û be Û the ŒŽŒŽŒ ¬Ü corresponding r . Let Û Û coordinate functions. Any local section of ŒŽŒŽcan be uniquely extended Œ Ü Û Û  to a section of such that

¦ î Ê M¦ ¦ ¦ Q KJJKJ4 Q   JKJJ4  Ê H’ JJKJH ’ Ê o ’ JJKJ ’ ` Ê • î @ 9¦   L œ L ` O

½ & i Q ¦L œ L ` ` Ê

 & î 9¦ â &À iQ ™ @ â Q L œ ‡ JKJK‡ J L â Q& iQ #J Lœ L L L Let be the space of sections of î Ê 9¦ (fields,œ and their partial derivatives). Roughly speaking a Lagrangian of order is a smooth map from î Ê 9¦ to the space of densities on  . We will be usually dealing with Lagrangians of the first order. Then we can write D





Û

݌ŽŒŽŒ

ŒŽŒŽŒ Ü

Ü

Û

Ü

r

39

LECTURE 4. THE DIRICHLET ACTION

40

@ 9 Q ð D  QÀð J Ö L L So, the action will be defined by a formula Ð  â  M& @  9Q & W & #J One can generalize the Euler-Lagrange equations to the higher-dimensional case: â â â F â â @ O } @ o~ KJJKJ4 Âà  J (4.1) L L Here we assume that the equality takes place when we evaluate the left-hand side on & . We also assume here that is of the first order. Let us consider an example, which will much relevant to the string theory. % © bebe very First, a little of linear algebra. Let

two vector spaces equipped with nondegenerate bilinear forms and , respectively. We can define a symmetric bilinear % © form on the space of linear maps Lin by ¿ & @ Tr M¿ & #

©  % is the adjoint map with respect to the bilinear forms and (i.e. where ¿ I


@ b ¿  ) for% any R % , R © ). Let© us explain this definition. Choose '¿À  # a basis

KJKJJ4  in and a basis JKJJ4 in . Let be the matrix of ˜ basis. Let be the matrix of in the first basis` and be the matrix of in` the second ¿ with@ respect to the bases, and is the same for & . Then the matrix of ¿ is equal to ` a so ¿ & @

@ v_à

Tr (4.2)   ` a a L _ @ v p @ à p @ # @ L anda we employ the physics where @  , where ` ¿ KJisJKJthe L  map summation notation. Assume that defined by ¿À  L L L @

. Then@ v ˜ L  . Similarly, is an element of the dual basis take & L @ % © É É <

<

` ˜ defined by &½ . Then we get . If we identify Lin ¿ & É L w © @ @ % , we getL ¿

& É É andL6L with L É L @ É J L L É L6L É % © From this we deduce that the matrix of the bilinear form on with respect to

is equal to the Kronecker the basis product of the matrices . The L4` defines an inner product on % . So, our inner product on % % 4` and© could matrix © be taken as the definition of the tensor product of the inner product on and on . Now we are ready to globalize. Let be a metric on  and be a metric on  . Define the Lagrangian (4.3)  '& @ ð D &Àð g D} @ Tr D & D & D} J a Lagrangian as

Þ





Û

Û

Þ

Û

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j





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F

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à

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à

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ß s

s 

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à

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F

ß s

s

Ê

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9

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32

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32

g

D}

32

Ê

is the volume form defined by the metric Here defined with respect to the metrics Ê and  . The corresponding action

D&

and the adjoint



of

D&

41 is

 D D~} @

(4.4)   M&  ð ½& ð g  is called the Dirichlet action.

and is given If is given in local coordinates Q KJJKJ4 Q by the matrix K

 J K J 4 J

` then (4.4) can be rewritten in the in local coordinates by the matrix ` form  š â & â & D JJKJ D J @

(4.5)   M&  ð ð œ â Q â Q Q Q ‰ h ` D~} @ ð š ð D Q  JKJJ D Q . Recall that This follows from (4.2) and the fact that % , where JKJKJ¥  is an the volume form on a vector space is equal to ‰ JJKJ hœ `   `  ` orthonormal basis. U

Ê



 

Ê





»ãâ

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j

k

Ê »ãâ

Ê

ä 

Û

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U

â j

Û{ä

2

k

Ã

Ã

Ê

U

Ã

Ã

Ã

Ã


ští w preserving the met-

Observe the following properties of the Dirichlet action:

(A1) (isometry invariance) For any diffeomorphism rics,

º

  M& @  bÉ M& ç (A2) (locality) if  is glued together from  and  along their boundaries, then g   M& @   `  '&Àð   g ` @ W œ is a h new metric (A3) (conformal scaling) if w on  , and-Â à  w is@t  does not change if and only if  thec J new action, then the action functional % @ ÙÛ= © be a linear map of inner product spaces© and
% w  % Proof. Let &
% w R , be an isometry of inner product spaces. Then, for any w R &½ w ) p @ r w p &  « @ w 4` M&  ™ « É J

@ This shows that M& & , hence 4  `

'& ) @ Tr M& & @ Tr M& & #J Tr ) M& `

& Applying this to the case when are the maps of the tangent spaces, this implies that ™ð6ð D &Àðð g D~} bÉ @ ðð D M& ð6ð g D}  J Property (A1) now follows from the stan<

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å

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j

9



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Ä($ç

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dard properties of integration of differential forms. Property (A2) is obvious from the definition of the action. Property (A3) follows easily from formula (4.5).

@

9Q  1

@ 1  with metric defined Ù W] @ 1   . Define a metric on  by

Example 4.1. Let with a local coordinate and by a matrix  s in the canonical basis in

L

LECTURE 4. THE DIRICHLET ACTION

42

ð ð o ð6ð g @ v . Let ¿ 
1  1  be defined by the vector function M¿ ` ] # JKJJ ¿  ] ) . Thenx v v v v ðð D ¿ ð6ð g @ ¿ w Í M¿À v ) ¿ w @ ð6ð ¿ ðwð o ð6ð ð6ð g J gx @ Ø D x@ £ ~ o

4 So, if we take , we obtain the action ž g` ò  ] g ] which ` we used in the previous lectures. g Â6à tj o , I do not know any geometric meaning of the Dirichlet action. HowIf ever, let us see that for a fixed metric on  one can always choose a metric on  such that the action acquires a very nice meaning. In fact, the metric is chosen to minimize the action. Let us consider the Dirichlet action as a function of and @ \ compute its variation in the direction at ‘ ÈÈÈ @ 'ž \ H ƒ š FÝž \ ™ ™ˆ ƒ

Ȑ @ O . Note that, for any invertible matrix and any square matrix of the where ƒ g same size, we have ð H ƒ ð @ ð Ið H ƒ Tr 4` ð Ið J Ê

g

ʐè

g

Ê

GS

Û{ä

j



Ê

Ê

Ê

S

S

Ê

Ê

Ê

Ê

S

Ê

Žé

Ê

Ê

$S

é A

n

n

Ëo

n

o

Gn

o

n

ð H ƒ ð œ @ ð Ið œ orH cƒ Tr 4` ™ #J h h Also H ƒ ` @ 4` FӃ 4` 4` J Let @ ââ & ââ & Q Q @ . The matrix is the matrix of the metric &  . It can be viewed as the and metric on the image of  under the map & (called the world-sheet). Then  ž  @  ð ð œ Tr 4` 4 D~}  (4.6) h and ƒ ‘ ÈÈÈÈ  M& @  * \ H \ \ H b 4 D~}  \ \ 4 D~} @  ð ƒ ð hœ Tr ƒ 4`  F  ð ð hœ Tr 4`  Thus

n

Ëo

hn

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$o

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Ën

opn

ä 

Û

»{â

o

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â

Û{ä



»{â

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Ê

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43

@  ð \ ð co Tr \ š Tr \  F Tr \ \  ) D~}  J ` ` `  hœ ` @

F \ \ . Then Set  4` `   ` ‘ ÈÈÈ M& @  ð \ ð û co Tr \ \ \ ) Tr \  F Tr \ \ 4 D~}  @ 4` 4` 4` 4` 4` ü  hœ Ȑ  \ û o \ \  D} J  ð ð hœ Tr  ` F c Tr ` ü  Since this must be zero for all possible , this implies that a critical metric \ satisfies F co \ Tr \`  @ O J (4.7) This implies ð ð ` 8 @ co Tr \4`  ð ð ` 8

@ Â6à  . Plugging this in formula(4.6) we get where (    M&Åç \ @tc  ð ð ` 8 ð ð h ÿ D}  J @£c , and the metric onh  is chosen to be critical, the We get a wonderful fact: if ( action has a simple geometric meaning. It is equal to the twice the area of the world sheet &Å k in the metric induced from the metric of  . Ê

Ê

S

S

S

Ê

Ê

Ê

Ê

Ê

S

é A

é

Ê

S

ʗÊ

ʗÊ

S

Ê

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S

Ê

ʦÊ

Ê

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S

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Ê

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j

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Ôê ê

Ê

$

In physics the latter action is called the Nambu-Goto action and the Dirichlet action is called the Brink-DiVecchia-Howe-Desse-Zumino action, or the Polyakov action for short.



ðð v L L  L L

Remark 4.1. In the case when the metric on is Lorentzian, we have to replace Ê with Ê . Also physicists use the metric to “lower the indices”. If  s is the matrix of  s s a metric  in a basis then for any vector the vector u s u is denoted by u . In this notation formula (4.5) can be rewritten as

ð…F ð

v   JKJKJ¥   L L `  L L L  M& @  ð ð â â & â â & D} J  hœ L Q Q  L Remark 4.2. The tensor Ù @ Ù M& @ F co

D Q D Q J L L L L is called the energy-momentum tensor. By (4.7) this tensor is equal to zero if and only if is a critical metric for the action  M&Åç š . Observe that Ù @ ð ð F co

D Q D Q @ ð ð o F (c D Q D Q J  hœ L L  hœ L L L L L L L L v

Ê

s

Ê

Ê

â

s

Ê

s

Ê

»

s

s Ê

s

ä[Û

ä[Û

s

Ê

s

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Ê

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s Ê

äqÛ

s

äqÛ

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Ê

Ê

s

s

s

LECTURE 4. THE DIRICHLET ACTION

44

Ù @ is trace-less@ if ( @tc .  1  1 and the metrics ¥ are the standard Euclidean metrics. Ù @ F co Tr 4 ™ D Q D Q J L L

In particular, Assume Then

Ê 



s

s

Let us write down the Euler-Lagrange equations for the Dirichlet action in the case when the metrics Ê and  are flat (i.e. Ê »ãâ and  are constant functions). We get the Û{ä equations

â â Q g &â Q @ O } @ o KJKJJ4 ™(SJ (4.8) @ 1  / @ 1 , and this is just the Laplace equation with In the case when  @ , its solutions are harmonic functions. In respect to the metric . When Ê »{â 

ä

»

Û{ä

â

Ê

Ê

Ê

»ãâ

»{â

S »{â

general, the Euler-Lagrange equation for the Lagrangian (4.3) can be written in an invariant form:

( D & @ O

@ Ù  & pÙ with respect D where ( is the covariant derivative of a section & of ¦ to the natural Riemannian connection defined on the bundle ¦ . @ 1 g , with coordinates iQ Q @ ] Q , and the special case when @  @ Consider @

~ o

7 o

 1 g becomes diag F . Take . Then the Lagrangian density ` @ * â â & g FP ââ & g , D ] D Q J Q ] Tr

&

Ê

Ê

»{â

à



2

Þ

â & â & @ J â g ] F â gQ O g g @ @ o) Notice the analogy with the Lagrangian for a harmonic oscillator (with › @ D D Q g FÎQ g D ] J ]g We can view &Å ] Q as the displacement of the particle located at position Q at time ] . The Euler-Lagrange equation for the scalar field &½ ] Q can be thought as the motion equation for infinitely many harmonic oscillators arranged at each point of the straight line.  If is the flat Lorentzian metric in 1 defined by the diagonal matrix diag * F o~ o~ KJJKJ Ko , the Euler-Lagrange equation for a scalar field with Dirichlet action is @& â â & @ F â â g & F Ž  ââ g & @ O J ]g  g Q g The Euler-Lagrange equation is

˜

Þ

Ê

ë

ä

ä

ä

ä

45 ë

The operator is called the D’Alembertian operator or relativistic Laplacian. A little more general, if we take the Lagrangian

@ co â & â &§FΛ g & g p

the Euler-Lagrange equation is the Klein-Gordon equation & H › g& @ OJ Þ

ä

ä

ë







& r
& £  

In many quantum field theories is a fibre bundle over and is a section. When is a g -bundle with some structure group g a map is called a classical field, otherwise is called a non-linear ì -field. For example, when is the trivial vector bundle of rank , a classical field is called a scalar field. Of course any map can be considered a section of a fibre bundle, the trivial bundle .

&


š    Ü  á& 

o

DQ 

Ù L L @ ¥•
bÙÛ

9¦ p ¦  M¿4Z @ - M¿ Z H ¿ MZ p

where ¿ is a local smooth function and Z is a local section of ¦ . It is clear that the difference of two connections is a morphism of vector bundles and thus a connection is a section of an affine bundle. Each connection defines the Lie -valued 2-form, the curvature form, Ì @ D H co * , @ Ž Ì D Q D Q J L L Here Ì is a local function on with values in Lie . We define the Lagrangian density L on by setting @ Ì ‚Ì J Here is the star-operator on the space of differential forms with values in a vector @ D}

bundle ¦ equipped with a metric . It is determined by the property

where is a natural bilinear form on the space of such forms (determined by the D} is the volume form defined by Riemannian metric on Ù and the metric on ¦ ) and  the metric on . The Euler-Lagrange equation for the gauge fields is the Yang-Mills equation: Ž â â Ì HV* Ì , @ O @ o~ JKJJ4 D J (4.9)  QL L L` L 

Example 4.2. An example of a classical field is a gauge field or a connection on a principal g -bundle over . It is defined by a 1-form u n on with values in the adjoint affine bundle Ad hg . In other words, it is a section of the bundle :à Ad Gg . For example when g GL , a gauge field is a map of vector bundles n End where is a smooth vector bundle of rank w over . It satisfies n

;F

F

bn

GF

n

°

n

n

Gg

s

n

s

¦Ã

s

í

¾

2

hg

Þ

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º

U

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s

n

s

s

ïÅ



º

î

LECTURE 4. THE DIRICHLET ACTION

46

There are two approaches to quantization in higher-dimensional QFT. First uses functional integrals which generalize the path integrals. D Consider a space of fields j Map on a -dimensional manifold . We assume that , where ( a “time factor”). For each we denote D D > !  be the space of fields by the restriction of a field to . Let D D  on obtained by restrictions of fields from . Fix two fields D Consider an action and set

@  @ Ù Ü  w  Â6à ٠@ o  / ( Ù R & &;R   @ ] Ü  w  ¿ R ] Ð ’ @ o~ c J  aa a   1 
U L  (4.10) M¿ ¿ g @ ¼ h  L 3 657 (+* &-, Ð ` @ ¿ . We use some where we integrate over the space of fields¼ œ & on  such that & measure (+* &N, on  . a L @ Observe the obvious analogy with our previous definition where we take  w point and  is the set of maps = . Now if we consider some Hilbert space of functions on  , the integral operator with kernel (4.10) defines a linear map a Ù
 Ù M¿ @  M¿   - G(+* , É        aœah aœ ah aœah
\  \ such that It also defines a self-adjoint Hamiltonian operator Ù @     L a h ba œ Ü  , a œ a h map which can be used to define a Hermitian

@ ð  aœ  ah J g ` g   L a h ba @ œ ` h The kernel  has special meaning for ¿ ¿ . The integral ¼ œ ¼ h Tr   @  ` g > (+* ¿b,  É   ¼  ¼   L a h ša œ  is the trace of the operator  the partition function of the theory.    L a #h ba @ œ o~. JKItJJ4is )called •— be a local quantum  field at a point   on ’ generally, let  (aMore local field is a functional on  which depends only on &Å and derivativesL of L L   & at ). An example '&½   ) , where À
<Úá1 of a local field is the functional &   is a function on . We set >    ç k @  ¼ 657   M& G(;* &N, (4.11) h 3  L ¼œ ¼h L L ¼œ L L L This leads to the correlation function >   (4.12)   JKJJ     @ ¼ œ  >¼ h  L kL ç k J ` ` ¼œ ¼h }



D

!

D

D







}

ið





ð





Map

.





@





ð

ð







@

ð

W

4ñ

ð



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@

Map

@

ò

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ôó

l



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ò

ò



ò

hò

ó

47

@  b É  w      @





ç] ]g g g M¿ g M¿ >    ç k G(+* ¿ , (+* ¿ g , J ` ` ` ` ¼œ ¼h `   œ it defines a linear operator h in . Thus This is still linear in and half-linear g such that `   ç ] ] g g @ g ð   L  a ša     L  a ba  J œ it go to infinity, h œ i.e.` define ` ` @ ] F ] by hletting We can get rid of the parameter ] ÁÂ6à [   ç g @ Á6`Âà [ ð   pJ g `   a Ä g   L a 7L a ` a Ä In this way we get a local operator Í   in the Hilbert space . It is called the vertex operator associated to a functional . Another approach to quantization generalizes the one we used for the harmonic @ Ù Ü  w . For any field &
<= we denote by oscillator. we assume that  â \ the partialAgainderivative in the time variable. By analogy with classical mechanics we We can use (4.11) to define a Hermitian form on the space Map }

/

hò

%õKö

/

£õbö



/

/



/

/

hò

}

/

hò

ò ÷

@

 /

/

D

iò



/

}

of functions on



@

 /

/

@

ò ÷

@

ò ÷

ò ÷

/

/

 ñ





ò

Þ

introduce the conjugate momentum field

u ] Q @ â \ M& pJ @ ) 5 g Fz 5W g we obtain uú@ 5 . For example, when '& introduce the Hamiltonian functional g` a :  a o Ø @ u D

J A '& c  É & F Q S

Þ

S

¶

¶

¶

¶

¶

Þ

¶

We can also

.

Then the Euler-Lagrange equation is equivalent to the Hamiltonian equations for fields

½& Ø ] Q @ u ] Q

uØ ] Q @ F

‘&½ ] Q

Sã.

Sã.

u

S

J &

S

where the dot means the derivative with respect to the time variable. Here we consider and as independent variables in the functional . and use the partial derivatives of .

u

&

&

To quantize the fields and we have to reinterpret them as Hermitian operators in some Hilbert space  which satisfy the commutator relations (remembering that is an analog of and is an analog of ).

ã

u

ãØ

* ] Q p ] - , @ .’  iQ§F - #

* ] Q # ] - , @ * ] Q # ] - , @ O J 0

Ž0

ø

Õ0







ø

S

$ø



(4.13)

(4.14)

48

LECTURE 4. THE DIRICHLET ACTION





}

Here we have to consider 0 ø as operator valued distributions, i.e. a continuous linear functionals on the space of test functions on equipped with some measure with values in the space of operators in a Hilbert space  . Any function on with values in the space of operators in  which is integrable with respect to some operator-valued 3ù measure defines a distribution

D}



 &   Ù iQ &½ iQ D} J 3ù

The commutator of two operator valued distributions is a bilinear form on the space of test functions:

'& Y * Ù ' & # )Ù g , J ` Thus the meaning of (4.13) is * M& # , @  &½ iQ 9Q D} J  @ 1 and is the Klein-Gordon Lagrangian 5 F W 5 F Example 4.3. Assume that  w hh › g & g . A solution of the Klein-Gordon equation can be written as a Fourier ha integral h º ] Q @ s oc‚u v  L  Ê W  a  H v  L   Ê W ‡ a  D

where Êg @ g H › g J Similarly we have a Fourier integral for ] Q :

] Q @ s ’c‘u  º Ê )F Ê W v H Ê W v D J  L  L a  L  L a v v v v To quantize we replace with an operator Ê and with the adjoint operator Ê and consider the above expansions as operator integrals. This implies that the operators

Q and ] Q are Hermitian. The commutator relations (4.13) will be satisfied if we ] require the commutator relations * v Ê v Ê É , @  IF w p

*v Ê v Ê É , @ *v Ê v Ê É , @ O J This is in complete analogy with the case of the harmonic oscillator, where we had v v satisfying * v v , @ o * v v , @ * v v , @ O (or • only one pair of operators

v v v v @  * v v , @ * v v , @ O ). There is a big operators

satisfying * , v v difference however. In our case the Heisenberg Lie algebra generated by o Ê Ê is L L L L L L infinite-dimensional. /

Ž0

/

$ø

/

/

Þ

¶

0

ž%ú

ir

˜

ž%ú

hr

¶

¶

¶

r

r

ø

ø

´û

˜

ir

ð

´û

ž ú

ir

ir

ž ú

r

Ó

hr

ø

Ó

S ir Ó

Ó

Ó

r

Ó

s

Ó

Ó

S

s

s

Ó

Ó

s

Ó

Ó

Ó

49

Exercises Þ Þ

@ D & be a Lagrangian on  with a metric defined on the space of PU1  . '& Define the energy-momentum tensor by Ù M& @ Ž ⠍ â M& F M& Ê

4.1 Let maps

Þ

Ê

»{â

â

»

ä

Þ

S

Ê

S

»ãâ

ä

Û

W …57 @ O if & satisfies the Euler-Lagrange equations. 4.2 Consider a system of ­ harmonic oscillators viewed as a finite set of masses v arranged on a segment * à , , each connected to the next one via springs of length ƒ . Write the Lagrangian Ö Mƒ describing this system. Show that the limit of Ö Mƒ when ƒ goes to zero is equal to  â &½ ] Q â &½ ] Q D J â ] g F g â Q g, Q x ` where are some positive constants and &½ ] Q is the function which measures the displacement ` g - of the particle located at position Q at time ] •. 4.3 Let be a Riemannian manifold of dimension and ¦ be a vector bundle over equipped with a Riemannian metric. Show that there exists a unique linear Ê Ù ¦  ®  @ Ê Ù } ¦ such that, for any R
® isomorphism  v ½ . à Here @ is defined Xr ® Ê Ù ¦ p one has v à.

locally by extending via linearity the product ` w w ÊÙ Ù Also v ` is à the inverse

w @ à metric

w Êon v àKextended

#J to  ®  ¦ by the  formula 4`     4` 4.4 Using the star-operator defined in the previous problem show that the Dirichlet action can be rewriten in the form  ž '& @  D & D &

D where & is considered as a section of the bundlle Ù K & ÙÛ . (i) Show that this definition agrees with the one given for the Dirichlet action;

(ii) Prove that u

¶

ãüqý ¶

ý

e

þ

þ

þ

þ

G¾



¾



;¾

I

¾



(à

¾

+à

&

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î



+à

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50

LECTURE 4. THE DIRICHLET ACTION

Lecture 5

Bosonic strings 

( d@ c  ` SÜ j 1 o 

From now on we stick with dimension of our QFT. This is where strings appear. Our manifold will be a smooth 2-manifold with a pseudo-Riemannian metric Ê . It could be the plane or a cylinder , or a torus , or a sphere , or a compact Riemann surface Î of genus  . Of course each time we should specify a metric on . † We shall begin with the case when is a cylinder (closed strings) or †

î (an open string). We use the coordinate in the circle direction and the coordinate (time) in the -direction. A map can be considered as a map Þ † †

1g



 @ 1 Ü *O , ]



1

`Ü ` `Ü 1

ž
|  &½ ] ™ #

g

&½ ]

&À ]
~1|  p ]  where  is the loop space of  , i.e. the space of smooth maps from a circle to  . In the case of open strings  must be replaced with the space åæ  of paths in  . We shall consider only closed strings, however occasionally we state the corresponding results for open strings. @ 1  with the Lorentzian flat metric in the beginning that  @ We shall @ also assume @

, where diag GF o Ko KJKJJ KoB . We will write vectors in    K

K J  J 4 J



K J  J  J



K

 J K J 4 J

 as 9Q by iQ Q the vector )FžQ ` Q general Q equal to g target Later on we will of course consider more spaces 9Q ` JKJKJ4 Q Q and . denote ` . We action ` consider the Dirichlet   ž '&Åç š @ Ù c  ð D &½ð g D~}  @ Ù c  ð ð â & â & D}  J (5.1) c‚u w for Here Ù is a certain parameter of a string (the string tension). It is equal to o7ˆ open strings, where w is a certain other constant called the Regge slope. For closed @ o7ˆ u w . We use the subscript to emphasize the dependence of the action strings Ù on . (5.2) ž &Åç š ) @ ž M&Åç b p



Þ

Þ





Gß

ß

Û{ä

ä



ä

ä

Û{ä

Û{ä



ä

„

Ê

Ê »ãâ

Ê

»

Û

â

ä

º

Ê

º

Ê

º

hº

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º



Ê

51

Ê

52

LECTURE 5. BOSONIC STRINGS



¿;RSï [ k

where º is a diffeomorphism of . This means that the action is invariant with respect to smooth reparametrizations of the maps. Also, for any , we have

ž M&Åç  ¼ š @ ž '&Åç š #J Ê

Ê

(5.3)

This means that the action is conformally invariant. It is known (see, for example, [Modern Geometry] by Dubrovin, Fomenko and Ê Novikov) that there exists a unique diffeomeorphism º such that º  Ê , where Ê is a smooth function and is a flat metric given locally by the diagonal matrix diag . By (5.2) and (5.3)

b @  ¼ \

\

¿ )F o~ oB

ž M&Åç š @ ž ¥& ç \ #

(5.4) @ D We shall fix the metric on  by equipping 1 with the metric F Q and taking  1°ˆ c‚u ‚m with the metric induced by the standard metric D ] g on g1 . Then we have ` two constraints on & . One comes from the Euler-Lagrange equation for the action of the energy-momentum tensor. Since the  and another comes from the vanishing Lagrangian function for the action  is equal to @ Ùc â & â & F â W & â W & (5.5) a a is the Euler-Lagrange equation for the action  â g F â Wg & @ O } @ o KJKJJ4 ™(SJ (5.6) a So our field '& satisfies the Klein-Gordon massless equation. The value of the energy-momentum tensor Ù at \ is equal to Ù \ @ Ù \ @ â & â W & @ ONç Ù \ \ @ Ù @ co â & â & H â W & â W & @ O J ` ` a ` ` a a (5.7) Ê

<

Gº

Ê

w

é A

é A

Þ

Û

Û

Û

Û

é A

Û

Û

Ê

Û

»ãâ

Û

Û

Û

Û

Û

@ ] H Q @ ] FÎQ J ‡  â â

Let denote the partial derivatives with respect to these coordinates. We have ‡  â @ cYo â W H â p â @ c—o â F â W #J ‡ a  a Thus we can rewrite (5.6) in the form â â & @ OJ ‡  This easily implies that a general solution of (??)EQ) can be written as as sum & @ & ‡ ¥H &  pJ To solve the wave equation (5.6) we introduce the light-cone coordinates ì

ì

Û

Û

‡ Gì Û

Û

~

Gì

53

5

& H &

c‚u

5ÿ

Using the boundary conditions, we see that, in the case of a closed string, the functions ƒ ‡ ~ , ¶
¸ÿ and ¶
¸ÿ are periodic with period w , so that we can use the Fourier Û Û ¶ ¶ expansion to write

õ

& ] Q @ 4co Q H \ ‡ H ’ Ž  \ • o    L 
õ 8

(5.8a) (5.8b) & ] Q @ co Q H \  H ’ Ž  \ • o    L 
ÿ 8

@ @ \ . We shall see in a moment a reason for the choice of and \ where the constant g . € ` Also, since we want & to be real,  @   @ J   The field & ] Q (resp. & ] Q describes the “left-moving” modes (resp. “right-moving” modes) of a closed string. Note that â & ] Q @ Ž  
8

(5.9a)    L õ ⠇ & ] Q @ Ž  
ÿ 8 J (5.9b)    L  It is clear that  Q @ \ g € & 9O Q D Q } @ o~ JKJJ4 ™( and can be interpreted as the center-of-mass coordinates. @ o QFT the momentum field is defined to be By analogy with (   @ â ââ @ Ù â & J & a a The expression @ Ù  \ g € D & D MO Q D Q @tc‚u Ù c \ @ c \ J (5.10) ] @ O . Now we can rewrite the is the total momentum coordinate of the string at ] equations (5.11) in the form & ] Q @ co Q H co g ‡ H ’ Ž  \ • o    L 
õ 8 (5.11a) & ] Q @ co Q H co g  H ’ Ž  \ • o    L 
ÿ 8 (5.11b) ‡

Û

~

ó

Q

º Û

ó

º Û ì

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ó

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ó

w

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w

º Û



ó º Û

Û

º

Û

z

z

LECTURE 5. BOSONIC STRINGS

54

@ 1 Ü   @ O ` ] Q Y @   bai‡ L WB98 J The Euler-Lagrange equation (3.7) gives âBâ  = b ~¯ @ O The equation of a string becomes & < @ c4o Q H c4o ’ g Á ÛH ’ ʎ  \ o Ê Ê

(5.12a) (5.12b) & <¯ @ co Q H co ’ g Á ¯ H ’ ʎ  \ o Ê ¯ Ê

can be rewritten in the new coordinates too. We have The stress-tensor Ù Ž  @ â  â  @ Ù  @ â  & â  & @ Ž

Ù J & & ¯ ˜ g ˜ g ˜ ֘  ˜ ֘ 

Remark 5.1. If we choose the Riemannian metric on instead of pseudo-Riemannian, we will be to identify the cylinder with the punctured complex plane   able !  W W by means of the transformation ô"

z

"

Z ¾

‡

"

Û

"

~

ó

Û

Û

m

¯

ó

Û

"

"

Û

¯

tm

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w

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ó

w

º Û

r



"

º Û

r



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»{â

"

Û

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Z Z

Z

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Z

Û

Û

"

{€

This is a familiar expression from the conformal field theory.

@  \ g € â &   F D Q @ Ù c  \ g € â & â & H â W & â W & D Q J (5.13) a a a Observe that it vanishes on a string which satisfies the constraint that the energymomentum tensor vanishes. Plugging in the expressions for & in terms of   , we obtain @ co Ž   H   (5.14)    Now it is clear the introduction of the constant . It made our formulas not depend on Ù . Observe that we could simplify the sum by getting rid of but we don’t do it, since in a moment the coefficients   will become operators. g` Using the new coordinates we can also rewrite the constraints (5.7) in the form â & â & @ co Ù \ \ HAÙ \ @ O

‡ ‡ ` â & â & @ co Ù \™\ F Ù \ @ O J   ` This immediately gives â M& â M& @ â '& â M& @ O J a a a a The Hamiltonian of our theory is equal to .

Þ

z

Û

{z

Û

Û

Û

Û

Û

Û

.

hº Û

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55

@ Ù c  \ € â '& ֘ Ù  a @ c \ € â M& ֘ a

@ O (5.15a)

â M & g ˜ W D Q @ co Ž   a⠍  L D @ o Ž  ˜  @

M& g ˜ W Q c  O J (5.15b)     L ˜ a Observe that @ \H \ (5.16) Ö Ö Now we quantize & as in the previous lecture by taking  as operators in some Hilbert space. Since we want & to be Hermitian we require  @   @  J   We need *   ] Q # ] Q w , @ Fž’  iQÍFÎQ w

This can be restated in terms of the Fourier coefficients as follows: ~

‡



~

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Û

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{€

‡

Û

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º

Û

º Û

{€

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Ó

Gº

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¸º

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4S

ä

ß

äqÛ

Plugging in the mode expansions, we see that this is equivalent to the following commutator relations

*Q , @ ’ (5.17a) *  , @ *  , @ ›  \

(5.17b) ˜‡ ˜ ˜ all other commutators between, Q   are equal to zero.

. In order they make sense We can also quantize the expressions for

Ö ˜ (byÖ ˜ analogy with the harmonic as operators in some Hilbert spaces we will require oscillator) that in our representation  kills any state provided that • is large enough.   Thus the sum  \ makes sense. Let us set, for any operators

with ˜ Ò , indices in an ordered set L

@  if ’ { ; (5.18) otherwise. L L L

, @ O,  It is called the normal order for the composition of operators. Since * @



˜ we can rewrite Ö ˜ › O in the @ formo Ž

֘ c  ˜     do not commute. Of course The situation with \ is more complicated since @  § • H  Ö     we know that so we can write o \ @ c \ \ H Ž \    H cYo ( F c Ž • Ö  ` ¯

Û

º

º

Û

º

ä

Û

º


ß

ä

º

Û

º

Û

Û{ä

XS

ä

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u

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n

n

s

n

n

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s

s n

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ä

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n

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º

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6

º

s

56

LECTURE 5. BOSONIC STRINGS

(} @ F c

o \ \O Ö Ö

}

Here we get because when we sum with respect to , the contributions corresponding to cancel each other. We shall deal with the last sum later. Now we define and  by dropping out the infinite sum.  Similarly we define the operators  . The operators are called the Vira  soro operators. Notice that and (resp. and ) are adjoint of each other.

֘ ֘ ֘ ֘ Ö˜ The expression of the Hamiltonian operator is now straightforward: @ \ H \ H ( F c Ž [ •—J (5.19) ‘ Ö Ö ` Since the last sum obviously does not make sense, we regularize it by setting [Ž • @ oB @ o J )F F o c ‘ ` So, finally we get @ \ H \ F ( o Fc c

(5.20) Ö Ö From now on \ @ co \ \ H Ž     Ö ` o Ž J \@ c \ \ H    Ö ` Let us find the commutator relations between the operators  . First we use the Ö following well-known identity: * ï ( , @ * ï, (zH ï * ( , HV* ï, ( H ï * ™( , J This gives * Ê  Ê  , @ Ê ~* Ê  , H Ê  * Ê , H ˜ ˜ ˜           @ * Ê  , Ê H  * Ê , Ê Ê  Ê H Ê Ê  H ˜       ˜   HV* ó ˜    J  ˜    i›úF ˜ Ê   Ê , 9›£F ˜ Ê   Ê  } ó       Here we skip the upper index . This easily implies *  , @ co Ž Ê  Ê  H 9›ŠF Ê  Ê pJ ֘ Ö Ê ˜ ‡ ˜ ‡ ֘ Ö˜ 

.

v



.

º Û º



nÑo

n

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¼

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º

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6

Û

n

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o

6

Û

¼

Û

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n

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¼

rS

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¬o

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57

› HA• @ O *  , @ i›£F •¥  if › Hݕ @ O J Ö Ö˜‡ @ O we haveÖ ˜ a problem For › H• since Ê ÀÊ is not defined. Since * Ê Ê , @ Ê F , we see that the difference   t @ c * , › \ H i› id ֘ Ö˜ Ö @ O, for some scalar 9› . Using the Jacobi identity, we find that, for Hݕ§H › • FΛ ¥H i›úF •¥ ¥H F •¥ 9› @ O J @ o and › @ F • F o gives Setting c c •ÍHVo7 @ •æH •¥ • F|F o •æHÔo7 oB J @ v › ” H=à › for some constants v /à . We will fix the constants This shows that I i› v v in some when we consider the representation of the Lie algebra generated by ˜ ˜ Hilbert space. IF •

Changing r to r

¤'

in the first sum we obtain for

'

u

{€ º Û

Ñr

º

º Û

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n

n

r

n

ir

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ir

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Û

Û

Exercises

M `

v R Ö @  L   v   L @ ’# i›£F •¥  . (i) Show that * 

, @ Ö Ö ˜ . Set Ö ‡ ˜ (ii) Let Vir Vect M   ` * v # àK @ * ) p

, , æ where is a bilinear form on Vect M . Show that this defines a structure of a ` Lie algebra on Vir if and only if satisfies æ * , ÅH æ * , ÅH æ * 7, @ O J @

. Show •— › @ O unless •+H › @ O and (iii) Let æ •— › v à that • — F •¥ @ v • ” æH= Ö à/•  forÖ ˜ some R . (iv) Prove that two bilinear forms and w define isomorphic Lie algebras if and only if æ •— F •¥ F w •— F •¥ is a linear function in • .   @ 5.2 Let * ] ] , be the algebra of Laurent polynomials in one variable. For any ] v  Ê Ê ] Ê R 4` *@ ] ] , let Res   ] ) @ v . Define a bilinear form on * ] ] , by 4` &½ ] # ] ™ Res4` #J 4` a

5.1 Let Vect be the Lie algebra of complex vector fields on the circle. Each field  U  U is given by a convergent series u XW . Let U  , where U  . {€

W

GF

;ß

F

ß

o

GF

ß

o

o

o

o

F

ß

v

o

ß

v

F

o

o

v

F

ß

o

n

XW

o

o

o

o

W

u

d

ÉW

d

U

U

W

LECTURE 5. BOSONIC STRINGS

58

&

&Å   H &½   ÅH &½   @ O J @ Der * ] ] , @ * ] ] , be the Lie algebra of derivations of (ii) Let  * ] ] , . Show that  ` is a Lie algebra respect to the Lie bracket `   D   v D   `@ a   D with   * D # D àp , * D D D   , &½   ™ #J ] ] ] ] @ @

p

K

7 o

(iii) Let Ö ˜ GF ] ˜ ‡Å` a O @ MO . Show thato *  , i›£F •¥  H o c i› ” FΛ   ‚J ֘ Ö Ö˜‡ ˜  (iv) Show that any central extension of the Lie algebra Vect M with one-dimensional ` center is isomorphic to the Lie algebra defined by the commutator relations as in (i) Show that

is skew-symmetric and satisfies d

d

}

hW

W

W

}

d

U

}

U

W

d

U

U

d

d

þ

S

(iii).

þ

Lecture 6

Fock space * v à , Ì 
 Ì ¿ ¿À * v à , @ v¿À ¿À àp ÝF ¿À Kà ¿À v @ Ù  ™ˆ Ò @ ~[  \ Ù   ) ˆ Ò

Û  v à F à @ v F * v /à , , v where v à R . where Ò is the ideal generated by elements v O for all /à R )   is For example, if  is a commutative Lie algebra (i.e. * /à , isomorphic to the symmetric algebra Sym   ), i.e. a free commutative algebra generated by the vector space  ( isomorphic to the polynomial algebra in variables indexed by a JJKJ

basis of  ). In general, Û  has a basis consisting of ordered products JJKJ Ê where  is an ordered basis of the vector space  .  œ  ` % 
L L  representation Recall that a linear of  in a vector space is a homomorphism of %

the Lie algebras   End , where the latter is equipped with a structure of a Lie @ PF . We say that % is a  -module. By definition algebra by setting * l, % of the envelopping algebra, this is equivalent to equipping with a structure of a left

module over   . This allows us to extend the terminology of the theory of modules over associative rings to modules over Lie algebras. An example of a linear representation is the adjoint representation ad
  v @ * Q v , . The fact that it is a linear End  defined by ad  iQ @ O . representation follows from the Jacobi identity * Q *  ,, HV* š B* b QN,, Hd* N B* Q ,6, v R  for any v R An ideal  in a Lie algebra  is a linear subspace such that * /à , à  and any  R  , or, equivalently, a submodule in the adjoiont representation. An example of an ideal in  is the commutator ideal *  ~, generated by the commutators * v à , v à R . A noncommutative Lie algebra without non-trivial ideals is called a

. Recall the construction Let  be a Lie algebra over a field with the Lie bracket of the envelopping algebra   . It is an associative algebra over which is universal with respect to homomorphisms  n of associative algebras such that . It is constructed as the quotient of the tensor algebra

à

`à

s



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s

ú

iÅ



n



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<

µo



n

"

"



simple Lie algebra. We will be mostly dealing with infinite-dimensional Lie algebras. An example of such an algebra is a Heisenberg algebra. It is characterised by the condition that its center (the set of elements commuting with all elements in the algebra) is onedimensional and coincides with the commutator. Let  be a Heisenberg algebra and let 59

LECTURE 6. FOCK SPACE

60

w

"

be a basis of its center  . We define a bilinear alternating form

/ on  by

* v à , @ v àK bJ @  ˆ  w is nondegenerate. For Its kernel is equal to  w , and the induced bilinear form on ¯   6  à  @dc ¯  and ¯  has a basis  KJKJJ4  Ê<ç  JKJKJ4  Ê example, if  is finite-dimensional, such that ` `  @ @ @       O o ’ J L  L  L L Thus  is completely determined by the commutator relations * , @  NJ (6.1)   L  L So all Heisenberg Lie algebras of the same dimension are isomorphic. If  is infinite-dimensional, we assume additionally that  is m -graded, i.e. @ 

  where each linear subspace   is finite-dimensional, and @  w f*    ,   J \ ˜ ‡˜ Let @  \    @  ! \   J  ‡  It follows that *  , @ *  , @ O ‡ ‡   and the bilinear form on ¯  restricts to a non-degenerate alternating bilinear form on

ÿ in each     . Thus we can choose a basis in  and a basis   in (6.1). Together  such@ that  is @ determined  L L with L L relations * b , * N , * N , @ O bythesethearecommutator õ ‡ as commutator called the Heisenberg relations.  Notice " L that  L @ \  " "

j

s

S s

r

s



s

s



iÅ

r

s "

S

{€

,

|

{€

"

"

{€

" $"

*

*

Ì

are maximal abelian Lie subalgebras of  . Consider a linear representation of one-dimensional linear space defined by

< oB @ v

‡

in the

oB @ O J v x x ‡ Here RtÌ is a fixed parameter of the representation. Now we can define a linear representation of the whole Lie algebra  by taking the induced representation: % v @ Ind $%  k
@  & ('  %  Ì J õ x an associative Recall that for any left module  over Ì õ -algebra the extension of  scalars of to a -algebra is a left -module *) defined as the quotient linear "



 #

o

=à

n

n

n

à

o

61

v v v w › ÝH Ù @ v
j

\ (6.2) ] L ]  L ]  L ] L ] * \ ’ O KJKJJ and any other pair of variables commutes. The Ì ] ] ] , -module corresponding to the representation  is the quotient algebra of Ì ` * ] \ g ] ] KJJKJ , modulo the ideal v ] x ] KJKJJ . Let us first describe the `induced % v as a g \ module generated by ] \ F K     ! *  J K J  J



K

 J K J J linear space. A monomial in Ì ` g ] JJKJ ] Êof. Fordegree ] ] ] ` , is called normally ordered if any monomial ] 4` K ] write ` g œ
] K ]
@ ] K ] L œ L JJKJ Ê and i’ L œ JKJJ L ’GÊ @ œ
  KJJKJ
 Ê  for some permutation where ER ¥Ê .` Usingg the relations (6.2),` we can write ` J ] L K ] L @
] L KK ] L
H normally ordered monomials of degree less than K ] and œ the aboveœ theK normal We call ordering decomposition of the monomial ]

write it as n.o.d ] . For example, the normal ordering decomposition L œ of ] L g ] ` ] ` is equal to Lœ ]L @ ]  g ] ` ] 4` ]  g ] ` ] ` H ]  g ] \ @
]  g ] ` ] K`

 J K J J of linear spaces L œ L ] 4` ]  g , ` . We have an% isomorphism @ (6.3) v Sym &  #J  The vector ð O
@ o do % v is called the vacuum vector. The structure of a &  -module on is given by ] K ]  ]  L K ]  L Ôo7 @ n.o.d ] K ]  ]  L K ]  L Ôo @ œ œ n.o.d KK  œ K ÿ ð O #J œ ] ] ] L ]L v %

œ definedœ by Note that carries a natural grading K @ ’ HdJKJJ7H ’GÊ

where R  and ð O @ ,‰ +O . L œ L ` L  L ,‰ + Ýdˆ7Ù Ù à R v R › R  Û ‡ ]L

v

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LECTURE 6. FOCK SPACE

62

% v * Q Q KJJKJ ,

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O

O @




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

. restrict to  M¦ to get a symmetric bilinear form sym . One can show that % v Ð

Ðj v vW   * L W ’ Q %

Remark 6.1. More explicitly the representation of  can be described as follows. We identify with the polynomial algebra W and assign to the operator ¶ , to the operator ¯ ¯ , and to " the ¶ s id and hence we get a representation obvioulsy scalar operator id. Then ¶ ¶ isomorphic to . 



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this inner product is non-degenerate if isomorphism we see that



is non-degenerate. Recalling the polarization

  KJ JJ JKJJ @ •—o – Ž
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  # (6.4)
 Lœ  L  œ   Lœ  œ  L  where the sum is taken with respect to all permutations of • letters. Here we identify Sym M¦ with the space of polynomials in a basis of ¦ . This defines a symmetric  L agreement we shall drop 10 bilinear form sym - on Sym 9¦ . Following the physics ` in this formula. A similar construction can be given for any hermitian bilinear form. In fact, if we choose a positive definite hermitian form on ¦ we can complete the tensor algebra Ù M¦ with respect to the corresponding norm and obtain a Hilbert space Ù M¦ . This space is called the Fock space associated to the unitary space ¦ . The completion of the subspace Sym 9¦ is called the bosonic Fock space. Similarly we can restrict ourselves with the exterior algebra ®Û 9¦ identified with the subspace of alternating tensors in Ù 9¦ . Its completion is called the fermionic Fock space. We will deal with it later. We apply construction of the Fock space to the Heisenberg algebra over by @  the, where  is equipped with a structure of a unitary space. taking ¦  the  Lie algebra  with a linear basis o~  )• R=m } @ o KJJKJ4 ™( Let us consider with Lie bracket defined by commutator relations (5.17b): *  , @ › Ê \

(6.5) ˜ ˜ ‡ Â6à   @ ( for all • @ O . Let Let  be the graded Heisenberg algebra over with @ }   # o~ JKJJ4 ™(S be a basis in   such that, for any •;j O , *  , @   bJ   ˜ ˜@ 2 @ 1 is viewed as an Consider the direct sum of Lie algebras  , where 2 

2 abelian Lie algebra. Let \ be a basis of . I claim that  is isomorphic to  . To see  sym



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   as follows. \ if • @ O ;  6 @ 35 L   if • @ O } @ O ; ¿À   4 (6.6) •if @ O , 57 ` \ @ o . It is clear that this is an isomorphism of Lie algebras. We call  the and ¿À  % v oscillator algebra of @ 1 . Let be the linear representation of the subalgebra  b



generated by  L ’ O described above. We can extend the representation to the whole  by setting @ ðO š \ ð O  1 \ R . Since  belongby to% thev ç center, for some @ the KJJKrepresentation.

, whereit defines J R 1 . ItWeis denote the obtained representation more natural to consider as a linear function on the ideal 2 of`  generated by \ ’s so @ ¥ \ . We@ will be interested only in representations corresponding to v @ o that % % o ç . Its vacuum state is denoted by ð . Recall that we can so that we set % write any element of as ð ƒ7 k
@ ƒ Ê   Ê Ê JKJJ Ê ð p

@ 9ƒ   KJJKJ  ™ œisœ a tensor œ symmetric œ  in lower and upper indices where ƒ   k

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defining a linear map  ` # finite-dimensional support. It is called  @ with œ 

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 R m \ and set a Lorentz polarization tensor. Fix 8  ð ƒ7 ¥ç#8 @ ƒB ç KJJKJ4  @ n ` n Ž n  ƒ Ê   Ê Ê JKJJ Ê ð pJ œœ  œ œ  ` ` %œ Let us define the inner product in . We may assume that ð is of norm 1. Recall that we want the operators and to be adjoint to each other. Then ð # ð @ ð p ðL @  L H :9 ð p ð @ :9 J L L  L   ð form L an othonormal  basis L in Minkowski So we see that  sense. In particular, \  ` ð  have squared norm equal to F • . Following the discussion above the vectors % we can extend the inner product to the whole . Two different monomials in ’s are orthogonal and L @ ðð Ê JKJJ Ê ð ðð JKJJ 

` œ  if all KJKJJ  are distinct. Weœ leave to the reader to deal with the general case. Note @ that, for ` any §RSm \ , and ƒ Mƒ KJJKJ ƒ R 1 , (6.7) ðð6ð ƒB ` ç ð6ð @ ð6𠃂ð6ð @ ƒ ƒ J 

this we define the linear map

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LECTURE 6. FOCK SPACE

64

¦

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Ë @ iQ - … 9 ] ¦  R+¦   ]

* ] ˜  ]  , @ ›Ó 

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Remark 6.2. One can define the Lie algebra   and the Fock space ic in a coordinatefree way. Let be a vector space over a field equipped with a non-degenerate symà metric bilinear form  . An element of  can be interpreted as a 9 à 9 finite linear combination of tensors . Consider the Lie algebra $" with generators 9 à , where " is central, satisfying the commutator relations

* ) Ë • ] R;, m ¦

  NJ (6.8) ˜ ˜  @ 1 is the Euclidean vector space, by choosing an@ orthonormal basis in ¦ , we If ¦ @  . One defines the Fock space 9¦ Sym ]  Ë * ] ¦ . Its@ see that … 9¦ 4` )• 4HS` JKJJMH • Ê elements are finite linear combinations of tensors  JKJJ  ] Ê •Ej O  JJKJ  R Sym

 ` by extending on M ¦ is defined 9¦ . The @ inner product   œ

*  the bilinear form on Ë ¦ to the symmetric  œ  ] ]˜ ] ] 4` defining ˜ product. The Lie algebra … 9¦ ˜ has a representation in 4 9` ¦ by to be ] ˜ j ˜ for and letting act by multiplication: the adjoint of › O  ˜ ]  ˜  JKJJ   @  ˜  ]  JK˜ JJ  J ˜ ˜ ] ] ]  ˜ \ Ôo   œ    ˜  œ   Also we let act by \ Ôo  JKJJ  ]  @ Å \ #

  
 œ where ¦ Ë is a fixed linear % form. It is easy to see that, in the case ¦ @ 1 , we get a representation isomorphic to . % There is one more important requirement on the spaces . The Lie algebra @ % 1  of the Poincar`e group å of the Minkowski space must have a linear ` 4  ` representation in these spaces. Recall that å is the semi-product % and the orhogonal group@ ( F o~ oB . The Lie algebra ofof å theis@ translation the direct group v R % 1 product of the abelian algebra and the algebra of matrices @ @ v v  1— satisfying H O . It has a set of generators  ’ o KJJKJ4 ™( 

o S (

» and satisfying the » commutator relations L L ’ * , @ O ’ @ o KJKJJ4 ™(S

(6.9a)  L *  Ê , @ ÊF Ê (6.9b) * Ê , @  Ê  L F ÊL  F L  H Ê  L    L  L     L  @ o L   @ o o (6.9c) @ o. Here corresponds to the matrices ¦ , where if ’ and F if ’ F ¦  Define L the operators L  L [ o Ž @ î Q FEQ FΒ ~ •   F   (6.10)   ` Then one checks that * î <; , @ Fž’ ; H ’ =; 9



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* î î ; , @ žF ’ ,; î H ’ =; î H ’ î =; FΒ î <; J   î is a representation of the This shows that the correspondence %  (one has only to replace the commutators Poincar´e Lie algebra in the Fock spaces * , with ’ * , ). Here the operators Q F oB Q correspond to the matrices 1 in the natural linear ( ~ o 

representation of of vector fields in L as the vector fields @ >  ? F @ Q BW A F×Q W . We can inviewtheaspace % as a polynomial function state ð ƒ7 Cç 8 R  

1

on Sym &  with values in Sym so that the vector field acts naturally on such states. The translation part of the Lie algebra of the Poincar´e group acts via the operators . % Remark 6.3. Recall that an irreducible linear representation of the Poincar´e group is described by the following data. First one restricts the representation to the translation % @ subgroup Ù . Since the latter is an abelian group the linear space decomposes into the @ E % D % D % #

F Ù ‚

direct sum of eigensubspaces R
]  F ] š HG ] R @ SO • R F o~, where B o

Ù Ù J The Lorentz % D group easy to see that the @ O is an orbit of acts. Leton be. Ittheisisotropy set F R Ù ;
subgroup of some F \ R . Then the restriction of the representation to defines an irreducible @ ˆ lifts to an action % @D . Now representation of in the natural action of on @ Ü Ü % % D D ˆ on the vector , where acts on the product by æ š < @ bundle  ¦  . There is a natural action of on the space XY M¦ of sections ` representation % is realized as an irreducible subrepresentation of this bundle and the

of XY M¦ . % For example, consider the irreducible representation which contains a vacuum Ù vector ð . The translation group acts via the operators \ . This shows that ð is an eigenvector corresponding to the character =RV 1 . This shows that the fibres @  • F o~ oB acts of the vector bundle ¦ are one-dimensional and the group @ •  o K

7 o —

 identically on the fibre over ÓR . Thus the data describing the  F representation consists of the orbit of determined by ð6ð ½ð6ð g (if the norm is positive @ SO • F oB is compact) and the trivial representation then the group SO • F o~ oB • ~ o 

B o

of SO F . Physicists say that ð transforms like a scalar. % We can define the similar space corresponding to right movers. Its vacuum % % . Its vacuum state is denoted by ð . Then we consider the tensor product ð . state is ð Its vectors look like this JKJJ ð

ƒ Ê   Ê é é       Ê JKJKJ Ê ð _  œœ JJKœ œJ þ þ and œ Zœ ˜ Z  JKJJ  œ Z œ and the þ _ þ polarization tensor where  ` ` ƒ @ 9ƒ Ê   Ê 4é` é     ` œ œœ þ þ } is symmetric in and 9 (resp. in andœ ) separately. % % One defines the norm on L similar to the norm on and then gets a non% %

degenerate inner product on . Finally we complete this space to get the a

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LECTURE 6. FOCK SPACE

66

@JI º % % pJ

Fock space of the closed bosonic string theory D

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% % ֘ @ o \ c \ \ H ­ \ @ co \ \ H ­

Ö Ö where ­ @ Ž   ­ @ Ž   J   ` ` @ Ž

@ Ž

› @ O     ֘  ˜ ֘  ˜ ` ` The operators ­ and ­ are called level operators. It is easy to check that ­  Ê JKJKJ Ê ð @ HÔJJKJBH  Ê JJKJ Ê ð ` œ  œœ  and a similar formulaœ holds for ­ .

 , @ i›ñF •¥  H i› id. We have to find the constant Recall that * @ v

= H à ֘‡ 9› › ” Ö * ˜ › . Ö One applies to some ground states to compute these constants. Notice , ֘ Ö˜ that @ c°o Ž   ð @ O › j O ð (6.11)  `/ ֘ \ ð @ co \ \ ð @ co ð6ð Åð6ð g ð (6.12) Ö Also @ co Ž @ co \ H \ ð @   ð ð ð #J  4`# Ö 4` ` 4` 4` Here we used that the operators and are adjoint to each other. Thus Ö Ö ˜ ˜  o Åð Ö Ö ð @ Ö ð p Ö ð @ ð6ð c g \ ð ð6ð g @ ð6ð ð ð6ð g @ ðð ½ðð g

` ` ` `  ` and we obtain Åð * Ö Ö ,“ð @ Åð Ö Ö @ ð6ð ½ð6ð g @ 4` ` ` ` ֏

Finally, let us see the representation of the Virasoro algebra generated by the operators (resp.  ) in the space hc (resp.  ic ). Recall that 

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67

½ð! c Ö \ H I o7 id ð @ ð6ð Åð6ð g orH oB g pJ @ v Heà @ O . Also This gives I oB ½ð Ö g Ö g ð @ ð6ð Ö g ð ð6ð g @ o ð6ð c g \ H ð ð6ð g @    ` 4` ð6ð g \ ð ðð g H o ð6ð ð ðð g @Vc ð6ð ½ðð g H co (SJ  4` ` Here we used (6.7) and ð6ð ð ðð g @ ½ð ð @ Åð H ð @ 4` ` H ` ` 4` ` @ c ` ` ` @tc 4` @Vc (SJ ½ð ð ½ð ð ` 4` ` 4` ` 4` ` 4` Thus Åð * Ö g Ö g ,“ð @ Åð Ö g Ö g ð @ ½ð Ö \ H c id ð @   c ð6ð Åð6ð g H c @Vc ð6ð ½ð6ð g H co (+J c L@ K v H c à @ ¤ v @ ( , hence v @ ( ˆ
• R m , we% have the followingg` commutator relation for the Virasoro operators for all › acting in the space . *  , @ i›úF •¥  H o o c ( 9› ” FΛ   J (6.13) ֘ Ö Ö˜‡ ˜  

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iQ -

 be a graded Heisenberg Lie algebra. Let Sp # be the symplectic 6.1 Let  group of linear automorphisms of  which preserve the alternating from . Construct a linear projective representaion of Sp & in the space M which is compatible with the representation of  in .  c . 6.2 Compute the norm of the state



% v Ö ” ð Ê JJKJ Ê ð . 6.3 Compute the norm of any state œ œ space of eigenvectors 6.4 Let ½ •¥ denote the dimension of the operator ­ ã @ of ~[ the \ level • Å •¥ 㠏 . Explain with eigenvalue . Compute the generating function Tr ë º Û

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68

LECTURE 6. FOCK SPACE

Lecture 7

Physical states for bosonic string The expression for the Hamiltonian provides the mass-squared formula . Recall that in the special relativity theory the mass is defined as the negative of the norm of the moment vector in the Minkowski space-time. Let us explain it. We use the metric in the space-time with coordinates defined by þ . Here , where is time and þ is a constant equal to the speed of light. To describe the motion we use the Lagrangian density

@ iQ \ ß D Q \g F D Q g F ÷ 1





\ i Q Q Q Q @ ” D Q g F D Q g” Q \ ] ] ` g ` g @ Fž› F|ð ß ð D ] @ › g y o F g D ]

g wg g @ @





ß 9Q Q g Q ” and › is a constant called the mass. The energy and where ß w the moment for this ` are equal to a Lagrangian @ â â Ø @ › g ˆ y o F g ’ @ o c Ÿ-

L Q g L ¦ @ L QØ F Ö @ › g ˆ y o F g J g L We have ¦ g F g ð Nkð g @ › g ÷

O so if we set @ M¦ ˆ g ˆ ‘ g ˆ ‚ ” ˆ K #

` O we obtain that @ F ð6ð ðð g

› g O O where we use the Minkowski norm defined by the matrix diag * F o~ Ko Ko Ko , . The vector @ JKJKJ4 so we can is called the total momentum vector. In our situation ` Þ

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LECTURE 7. PHYSICAL STATES FOR BOSONIC STRING

70

define the quantum mass-square operator by

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@ F \ \ dorHÔo \ \ #J @g K g so that  We shall scale the masses to assume that  g @ GF \ H ­ ÔorHdo )F \ H ­ #J Ö ð isÖ equal to Thus the mass-square of the ground state ð F ð Åð g @ g F gg HÔJJKJ F g J ` had Virasoro constraints Ù @ O . It follows Recall that in the pre-quantized theory we from (5.15) that the analogs of these constraints in the quantum string theory are the @ conditions that element of the Fock space. However, because @ O . Thus we have to require *   , @ @c •Ö ˜ \ H id,O forthisanywould imply that @ v Ö Ö Ö  O onlyÖ for positive › and Ö \ that for some 0 and similarly for the ˜ Ö . We set operators ˜ @ R ž
@ O › j O \ F v @ O (7.1a)

@ R k
Ö ˜ º @ O › j O Ö \ F v @ O (7.1b) @ Ö ˜   Ö pJ (7.1c) A state satisfying these conditions is called physical. Also for any state and a physical state & , we have &Àð Ö  F @ Ö  '& ð F @ O •Îj O  with the sum @  \  belongs to the nullThus the intersection of Ö  states. The Hilbert space . The elements of this space are called spurious space of which we want will be the quotient @ ˆ J ‡ in (6.9) commute with Virasoro operators, so that Note that the operators î defined the Poincar´e Lie algebra acts in the spaces of physical states. Remark 7.1. An abstract Lie algebra is called a Virasoro algebra if it can be defined by generators N  )• R m with commutator relations *  , @ 9›£F •¥  H › ” o FÎc ›   N q* N  , @ O J ˜ ˜‡ ˜  It can be shown that any any Lie algebra obtained as a central extension with onedimensional center of the algebra of vector fields on a circle is isomorphic to a Vira% soro algebra. A representation of the Virasoro algebra in a vector space is called a v representation with highest weight and charge if acts as a scalar operator id « and there exists a vector \ (called a highest weight vector) such that \ @ O › j O \ \ @ v \J ˜ ¯

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A universal representation with this property is called a Verma module and is denoted $þ . It can be constructed by using a similar construction as the representations by we constructed for a Heisenberg algebra. One considers the subalgebra Vir generated by the operators , then defines a one-dimensional representation by and finally takes the induced representation  Vir $à Vir 9 with positive W . Its elements are linear combinations of monomials ú ’s. Any irreducible representation with highest weight and charge þ is isomorphic D $þ to a quotient of . So, we see that each nonzero / generates a phys ic representation space ³ for the Virasoro algebra with highest weight and charge þ . Its highest weight vector is / . As we have seen before any physical state F QP belongs to ³ .

{ …o @ O › Ö j ˜ O › O % v % K

‡

Û JKJKJ \ Ö v  œ Ö   R v

\ with eigenvalue v , we obtain the Ö  g @ F× c \ F c ­ @tc ­ýF c v J (7.2) Ö Let us see which ground states in are physical. Since ­eð @ O Ö ˜  ð @ O › j O Ö \ ð @ co ð Åð g

and @ the same is true for the right mode operators  , we see that the ground state ð ð ð is physical if and only if Ö ð ½ð g @tc v J (7.3) For this vacuum state  gð @ F cv v We shall see from the next discussion that must be equal to o . Thus the vacuum vecSince all physical states are eigenvalues of mass-formula for physical states:

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tors have negative mass. Such states are called tachyons (they travel faster than light!). The existence of such states will force us to abandon bosonic strings and consider superstrings. D Let us look for / of level 1. Each such / has the form º Û c  . We phys ic Û have



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ð 4` @ \ @ ƒ 0

@ O › jVo \ @ co ð ½ð g HÔo7 Ö ` ` ֘ Ö Thus is physical if and only if (7.4) ƒ @ O ð ½ð g @tc v F c J @ 9O o~ O KJJKJ O , and then ƒ @ o O JKJJ4 O satisfies v If j o , we may choose @ @ o (7.4) but ð6ð ðð 𠃂ð g F . This means that we have ghosts, i.e. states of negative norm. This should be avoided since the the quantum mechanics deals only with Hilbert v o. spaces with unitary inner product. This forces us to take /

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LECTURE 7. PHYSICAL STATES FOR BOSONIC STRING

72

v o , we may take @ o~ O KJJKJ O and ƒ @ MO ƒ KJKJJ ƒ so we have ( F o -dimensional space of physical states of positive norm` and no states of nonpositive norm. v @ o , wec may take @ o F o O JKJJš O and hence ƒ \ @ ƒ . This shows that If we have a ( F -dimensional space of states of positive norm and ` a one-dimensional @ space of states of norm 0. The state is spurious and is physical if ð @ @ @ v space of ð ½ð g O . Thus, if o ,Ö 4` contains ` a one-dimensional c spurious states of norm 0. Factoring this space out we get a ( F -dimensional space ˆ , eachv element of which can be represented by a state of o positive norm. So far, we find that and no restriction on ( appears. @ R \ @ v # whereo R \ # R \ @ v pJ Let F o \ Applying and applying L we see that . LÖ Ö thatL ­ weL see@ L ­ that .Ö Hence Ö This implies L L F @ ƒ L   éL   K J  J J  J K J J #

ð Ê œ   Ê é _ œ   _ þ œÊ Ê œ_ _  states in HdJJKJBH  @ œ Z HÔJœ JKJ‚H þZ .  Letœ us look at  theœ physical where of level 1, ` i.e. ­ @ ­ ` @ . They˜ are of the form @ ƒ ð pJ 4` ` We have @ \ 9ƒ @ O › jtoJ (7.5) ™ @ ƒ #

ð ð Ö` ` 4` ` 4` Ö ˜ Similarly, @ ƒ @ O › j o~J #

ð (7.6) Ö Ö ˜ @ @ ð `½ð HÔo7 0 sothat` Also \ Ö Ö g` g ð ½ð g @Vc v F c J (7.7) In view of (7.5) and (7.6), we get ƒ @ ƒ @ OJ If

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@ ƒ ƒ @ ƒ ƒ J @ O , we have no restriction on ƒ L andL hence we have physical states of negative If @ O . If v o , we may assume that @ o~ O JKJKJ O so that ƒ \™\ @ O norm. So @ O for ’ @ O , we get a state of guarantees that is physical. But if we take ƒ negative norm, a ghost. So v @ oJ L @ @

, and then the condition is ƒ \ @ If O JKJJšis O equal @’ Oassume

ƒ ƒ \ O@ , weƒ may

KJJKJ4 ™( thatF o so that o F theo norm to ƒ g . We seeL that  g L` L ` L The norm of the state

is equal to

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73

if all physical states are of nonnegative norm. The states of zero norm satisfy 9 Û{ä .    º c c º Û º Û c ß º Û c c º Û Note that the states and  ä Û ä Û ä Û ä are spurious. Since c , these states are also physical because c c ß . It is easy to see now that any physical state of norm 0 u u Û and we can Û Û Û factor is spurious it out. Thus we obtain that for any non-zero light-like c D closed hc the space phys is of dimension and all its elements ¡ can be represented by physical states of positive norm. For any c of norm 0, the space of solutions of (7.5) and (7.6) is the Û{ä direct sum of one-dimensional space of matrices with nonzero trace, the -dimensional space of trace-less symmetric matrices and -dimensional space of antisymmetric matrices. The corresponding physical states are called dilatons, / ). gravitons and anti-symmetric tensors. These are massless particles (i.e. D Let us go to the second level, i.e. consider the physical states in phys ic of the form

@ ð c @ ƒ O 4 ` ` ð Ö ` 4 ` 4` ` ( g F c ( F c ( @ ( ( F ƒ @ 9ƒ ( ( oB o ( ( F Ÿ g` F F g`  g @ O @ ƒ ð ½H g ð p

@ ƒ . We have 4` `  where ƒ L L @ \ H

@dc 9ƒ H ð @ O

Ö` ` ` g ` o  ð ƒ @ \ H c @ 9ƒ H c ð @ O

Ög g ` ` 4` o o \ @ c°ð ½ð g ÎH c@ crð ½ð g H c J Ö ƒ@ c v ð Ö ð ½ ð ð Åð g ƒ @ ð ½ð g g ` @ O 4` F

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(7.8) ð ½ð g @tc v F @ F c J c R , where The norm of this state is equal to R @ ƒ ƒ H @ ƒ \ g \ F c Ž \ ƒ \g H Ž  \ ƒ g F \g H Ž \ g J @ ‚ O KJKJJ4 O # g @zc . Using Choose a system of coordinates in 1 such that (7.8) we can eliminate ’s and ƒ \™\ so that LR @ o ûŽ F c S  ` ƒ g H Ž 4 ` ƒ g J (7.9) L ` L6L ü L ` L This implies

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LECTURE 7. PHYSICAL STATES FOR BOSONIC STRING

74

Applying the Cauchy-Schwarz inequality, we obtain

{ Ž g ¹ ƒ g F ( c F S o Ž ` ƒ g @ Ž g ¹ ƒ g F ( c F S c ¤ Ž 4` ƒ g J (7.10)  L6L     L  L6L  L ` L ` norm. L c ¤ ,` all states are ofL non-negative L If (ýj c ¤ , theL ` state c withc S ƒ @ Thus, if ( @ @ @ o~ ’ {to and ƒ O ’ O ’ has the norm equal to ( F o7 GF (VH ¤ ™ˆ L6L O . Now if we take a physical state from of level 2 and of positive norm, L @c then

is an element of of negative norm. So we have ghosts. If ( ¤ ,a @ O ’ { o~ ’ @ . The states state of zero norm must satisfy ƒ ) ½H ð L @ 9ƒ H c 9 ƒ 6 ð ð ð6ð ¥H ð Ö 4` ` Ög g 4` `  are spurious. It is easy to see that any norm 0 physical state is equal to a physical spurious state. So we can factor them out c ¤ . and obtain only the space with only postive norms. Thus we have shown that ( @Šc ¤ consists of analyzing states of the next level. We skip it. The proof that ( v @ o~ ( @ c ¤ is called The result that does not contain ghosts if and only if the No Ghost Theorem. It was proven by R. Brower, P. Goddard and C. Thorn. v @ o and ( @Uc ¤ agrees with the definition of[ the Hamiltonian Observe that @ \ H @ \ F ÷ g using the regularization of the sum ‘ • . So physical operator Ö Ö O . g states satisfy ` invariant with Remark 7.2. One can show that one can choose a subspace in c respect to SO such that its states represent all states of positive norm modulo spurious states. This is achieved by a “light-cone gauge” which consists of fixing the first c S and the last coordinate & of the string. The group SO o acts in the space in via its induced representation. Thus defines a linear representation of the group parts invariant and hence defines a finite di c . It also leaves the homogeneous ‡ mensional representation in each space of given level. Elements of this space which belong to an irreducible component are interpreted as elementary particles. For examc ple, the anti-symmetric tensors of level 1 define the adjoint representation of SO . The dilatons define the trivial representation and gravitons define the standard repre÷ sentation of SO on the space  g 1 g . 

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7.1 Find physical states of level 2 in the Fock space of a closed bosonic string. D 7.2 By analyzing physical states of level 3 in ic finish the proof of the No Ghost Theorem.

Lecture 8

BRST-cohomology We shall discuss another approach to defining physical statesd which is called the BRSTD quantization. In this approach one introduces an operator in a Fock space of a d given string theory such that and D

@ g O@ phys

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We shall start with reminding the definition of the cohomology group of a Lie D   +à be algebra  with coefficients in its linear representation . Let  the tensor product of the exterior algebra of the space  and the space . This will D be an analog of our Fock space. If  Lie Gg for some Lie group g , then can ‡ be identified with the space À inv of left-invariant smooth differential forms on g with values in the trivial vector bundle defined by the space . Let be a basis of  and D

be the dual basis of   . Let so that .     £à Elements of are linear combinations of the decomposable tensors 

@

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® B  \ ‘   ï '¿ ï L  % ï L Ð ¿ JKJJ ¿ š ’ JKJJ ’ Ê R % Ð L v L @ @ ¿ and` to the contraction operator v@ @UT Let us assign to ¿ theœ operator T  J K J J  such that M¿ L ¿ @ L Ð GF L o7 Ê ‡Å` ¿  L JKJvJ ¿ JKJKJ ¿ if ’ L ’“Ê for some and O otherwise. product with the Koperator L œD L Jœ Q L L isarethelocalexterior L coordinates v IfL @  W Lie. Here  J K J the differential Q and on Q à

corresponding to the basis . We have ` L L L@ v v

v v @ v v @ O J L L L , L Here, for any associative algebra and Q R Q @ Q H Q It is called the anti-commutator or the Poisson bracket. The structure of  is determined by the constants Ê such that * Ê, @ Ê Ê J L L  L  75 à 9

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LECTURE 8. BRST-COHOMOLOGY

76

Ë @ š  . Define the BRST-operator in by L L @ Ž v Ë F co Ž Ê v v v Ê Ôo R End % pJ L L  Ê L L L L Lemma 8.1. @ O g @ v Ë , @ Ê v v v Ê to , where we skip the summation sign. Proof. Let We have ` L L @ vg v L L v v g` L Ë L Ë H L Ë Ë L @ Ž v v MË Ë FÓË Ë @ Ž v v Ë J ! ! L L L_ _ L L L L Also @ F co v v v Ê v H v v Ê v v Ê Ë J H L a v v v Ê v a H L v v Ê v a v @L g g ` ` @ ’. Using the anti-commutator relations we see that O unless ] @ @ v v v v v v v v v v v v v v v v Ê H Ê Ê L H a a Ê L so that In the latter case L L @ F L co Ê v v L Ê H Ê v Ê v L L Ë @ L L F Ž v v Ê Ê Ë J H L L ` g g ` @ L v v @ v v L L! Here we used that Ê F Ê and @ Ê F Ê . This shows that g H g H @ O . It remainsL to show O . We have `v v ` g g ` @ ¿ Ê ¿ v v v Ê v v v L that g @ v v v v v v v v v v Ê d o H ¿ ¿ _ ˜ L Ê _ a ˜ _ a ˜ L Ê do~J g@ g L _ ˜ @ a L _ a ˜ a is equal to zero. So If Z ] › ’ , the expression in theL bracket gg @ ¿ L Ê ¿ Ê ˜ a v L v v Ê v Ê v a v ˜ H v Ê v a v ˜ v L v v Ê do7H ¿ Ê ¿ _ ˜ Ê v L v v Ê v _ v Ê v ˜ H v _ v Ê v ˜ v L v v Ê do7H H ¿ LÊ ¿ ˜ v ˜ v v Ê v _ v v H v _ v v v ˜ v v Ê do7H a˜ a˜ ˜ _a v v v v v ¿ Ê ˜ ¿ _ ˜ L ˜ Ê _ a v ˜ H v _ v a v ˜ v L v ˜ v Ê do~J L a Using the Jacoby identity ¿ ˜ ¿ ˜ Ê H ¿ ˜ Ê ¿ ˜ H ¿ ʘ ¿ ˜ @ O

it is easy to see that each of L the four sums isL equalL to zero. D

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Applying the previous lemma we can define the cohomology of the Lie algebra  with coefficients in as follows:

 % @ Ker ˆ Im #J Example 8.1. Let  be an abelianÐ Lie algebra of dimension • . Its linear representation @ * JKJJ  , . Let % @ ï [ 1  with the action of  is a module over@   Ë
¿   ` ¼ . Then• ï   % can be identified with the space defined by   of smooth differential L L forms of degree @ ¿ a iQ D Q JKJJ D Q J Lœ L Lœ L The BRST-operator @ Ž v Ë L L D L D g @ O and coincides with the exterior derivative . We know that    % @  1  @ O ™•;j O J @ Lie . Assume that is a complex semi-simple group and let Example 8.2. Let  @% be its trivial representation. Then    @  #

where is considered as a smooth manifold. % In our situation we want to take for  the Virasoro algebra Vir and its represen

tation in the Fock space of bosonic string. The space ®  is called the space of ghost fields. We will be dealing with a version of the BRST complex which uses the semi-infinite JKJKJ ¿ we consider semi-infinite cohomology. Instead of differential forms ¿ @



K J  J J

\ forms. Let Ò decreasing sequence L L of integers such that \ 9’ and’ ` m ! \  be MÒ anym strictly ! \ are finite. A semi-infinite form is a formal m the sets Ò expression of the form @ ¿ ¿ JKJKJ¥J  @ ³ Ò m \ F ³ m n \  L 9Ò L mœ n \ is called the degree of  . The number ­ Let @ 

ë @ 9­ ­ F o~ JKJKJ #J Its degree is equal to ­ . We extend the operators v Ê and where Ò v Ê to semi-definite forms in the obvious manner. Note that any form of degree ­ can v Ê v  . Also observe that be obtained from ë by applying operators v Ê @ O j 0

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78

LECTURE 8. BRST-COHOMOLOGY

vÊ @ O

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‰V+  [ 8 g 9­ ­ * w @

i Q ' ¿ ¿À Q , w M¿ L @ ¿ * L  , @ ¿ ) i›úF •¥  @ ad w  M¿ ‡˜ L ˜ @ L ˜ L@ i›ŠF •¥ ¿  ˜ 9›ŠF •¥   ˜ i›úF •¥ ˜   J L ‡ L ‡ L This shows that @ 9’ÅF c •¥ ¿  adw  M¿ (8.2) L L @ O , we can set If •   ¿ ¿ JKJKJ @ ʎ \ ¿ JKJJ š  '¿ JJKJ @ L Lœ L L Ž ¿ JKJKJ ¿ ÿ 9’¥F c •¥ M¿  ¿ JKJJJ Ê \ L L œ @ L  Lõœ @ O . We • Observe that the sum is finite because . However it is not defined for • O @ easily check that, for •— › ™•ÍH › O, *   p  , @ • FΛ š  #J ˜ ‡˜ So our problem is to define š \ such that all Virasoro commutators work. Next observe that @ Ž i’½F c •¥ v  v @ Ž F •¥ v Ê v  Ê @ Ž • F v  Ê v Ê J   L L Ê Ê ‡ ‡ L We use this formula to set @ Ž
vÊvÊ
š \ Ê JKJJ Ê
denotes the normal order of a composition of operators defined by Here
putting on ` the right the operator annihilating the vector ë and inserting the sign of the permutation which has been made. If no such operators occurs among the factors v v @ v Ê v ʞF o so changing v Ê v Ê to F v Ê v Ê differs from we do nothing. Note that F Ê Ê the usual product. It is easy to see now that each   is well-defined. Let us compute * š  # š , . We have *   # š ˜ , @ • FE’ i›£F < p* v  v v v , J ˜ ‡L L ˜‡ r



Let Vir be the abstract Virasoro algebra with generators ó . We want to construct the representaion of Vir on the space of semi-infinite forms À of degree . Let ad  be the coadjoint representation of  on   . It is defined by ad . ó Let us identify with the dual basis of Vir. We have ó

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@ O , ’ @ › H  @ • H ’ . Then, its is easy to see that *!
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, @ v  v v v F v v v  v @ ‡ L L v ˜v ‡ H v v v  ‡ v L L @ L v  v L J ‡ L L L ‡L L L L ‡L Similarly we get *
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• j L Assume › O and Fž› O . Then *   # š  , @ c • F < GF • F < *
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, J ­ we have  ‡  ‡ Now, for any ’ such that ’ j ­

* v v v v , @ v o F v v v F v o F v v v @ v v F v v @ o7H
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v v
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@  ‡  ‡ ˜ ë! n ë ‡ c • š \ F oŸ~• ” F| 9­ g Fέ H • #J ¤ ¤ Finally *   # š , @ • FE›   H F oB¤ Ÿ • ” H M­ g H ­ H •¤   J (8.3) ˜ ‡ ˜ @ o ) the central charge is equal ˜ to . Also, If we fix the vacuum state (i.e. take ­ @ . Recall that` the representation of Vir in has the charge g` g¸ and the  \ ð vacuumvector ` ð ` is the eigenvector of Ö \ with eigenvalue v @ o . [ 8 ` g 9­ by We define the BRST operator (Bechi-Rouet-Stora-Tyutin) on g o @ v   F c • FE› k
v  v ˜ v 
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LECTURE 8. BRST-COHOMOLOGY

80

( d@ c ¤ , then g @ O . @   . We have Proof. Let   @g o Ž v    v Ê  Ê H v Ê  Ê v    #J g Ê !š d

Theorem 8.1. If ó

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LECTURE 8. BRST-COHOMOLOGY

82

Ž   •¥ 㠏 @ &½ ã J   We conclude that @  GF co ð w²w ð g oB ʎ \  g ÷  ™ co ð w ð g H c1S #J Let \ be the vacuum vector of  GF ð w ð oB . Set go ` g Ù @ Ê \  ÷  \
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 bç %
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8.1 Show that the equivalent definition of the cohomology of a Lie algebra  with coefficients in a linear representation   can be given as follows. Let be  the space of anti-symmetric -multilinear maps from  with coefficients in . Define   the coboundary map S by the formula


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