Field theoretical origin of excitation path integrals

Physics Letters B 275 (1992) 403-409 North-Holland

P H ¥ SIC S L EYYER $ B

Field theoretical origin of excitation path integrals ¢r A.I. Karanikas Nuclear and Particle Physics Division, Department of Physics, University of A thens, Panepistimiopolis, GR-15771 Athens, Greece

and C.N. Ktorides Center for Theoretical Physics, Laboratory for Nuclear Science, and Department of Physics, Massachusetts Institute of Technology, Cambridge, MA 02139, USA

Received 29 July 1991

Path integrals, describing quantum properties of elementary excitations, are derived directly from field theoretical models through elimination of the field degrees of freedom. Overwhelming emphasis is placed on spinorial field systems.

One of the most fundamental implications o f the second quantization process is the association o f elementary entities with corresponding field excitations. Particles and fields have assumed a unified existence in our intuition, to the point that field theoretical expressions are most often interpreted in terms of particles. A typical example is furnished by the Feynman diagrams which are looked upon as describing the propagation of (real or virtual) particles even though they actually depict perturbative expansions in field theory. A direct approach to the quantization of Ilarticle-like excitations has been described by Polyakov [ 1 ]. A path integral is constructed which refers to the euclidean space-time coordinates of the excitation. The various paths connect, in all possible ways, two points x and x ' in the euclidean space-time and they can be topologically nontrivial (e.g. self-intersecting in two dimensions, knotted in three dimensions, etc. ). In this note, we shall explore the field theoretical origins of Polyakov's path integral for point-like excitations. We focus our attention, for the most part, on gauge systems with fermionic matter fields. The invariant integration measure for the excitation path integral will be realized in a natural and straightforward manner. In particular, we shall sidestep any discussion concerning reparametrizations and functional metrics [ 1 ]. The impact of gauge interactions on spin factors will also be determined. Most important for us, however, will be the application of our recently proposed [2 ] non-perturbative regularization scheme for fermionic field theories. We hope that the effectiveness of our non-perturbative approach to fermions will emerge as a major by-product of this work. Our basic idea will first be illustrated, through formal considerations, for the simple case of a free, scalar field theory in d (euclidean) dimensions. We elect to work with the partition function, given the wider implications that it has in field theoretical descriptions. The resulting functional integral for the excitations will, consequently, involve closed paths. The propagator (open paths) will be presented as a special case. We start with the following, formal, representation of the free energy for a Klein-Gordon field theory in d space-time dimensions ( Z is the partition function) This work is supported in part by funds provided by the US Department of Energy (DOE) under contract #DE-AC02-76ER03069. Permanent address: Nuclear and Particle Physics Division, Department of Physics, University of Athens, Panepistimiopolis, GR- 15771 Athens, Greece. 0370-2693/92/$ 05.00 © 1992 Elsevier Science Publishers B.V. All rights reserved.

403

Volume 275, number 3,4

-lnZ=+½

PHYSICS LETTERS B

30 January 1992

Tr|_dTexp[T(OZ_m2)]7 3 T

Trln(-0Z+m2)=-½

(1)

0

For subsequent purposes we introduce the dimensional (length) parameter { and redefine define the mass parameter rno= {m 2. Formulating the trace in configuration space, we write Trln(-02+rn2)=-

i ~aT- e x p ( - T r n o )

fddx

°

T~ T/{. Let us also

(x°iexp(TeO2)]x°).

(2)

0

All one has to do now is to represent the matrix element ( x ° I e x p (Te 02)Ix °) as a path integral (over closed paths, of course ) in the usual manner. One obtains T

oo

Tr ln( - 0 2 + m2) = (const.)

1

~x(t) exp(- ~e f dt2(t)2),

e x p ( - Tmo) 0

.x{0)=x(T)

(3)

0

with the functional integral being defined as 1 r x(0) =x(T)

2(t)2)lim

f ( ) ~ _ _ , ,1U o d X ~N ) a ( x O _ xNN ) e x p l

(Xi_X*-,]2],

0

(4)

TIN.

where c~=-Had we chosen to work with the propagator, the corresponding parametric representation would have read C~q

G(x, x')= (xla2-£~m21x')=

_f d T e x p ( - m o T )

(xl exp(T{~02) [ x ' ) •

(5)

0

The rule connecting the two parametric representations is simple and intuitively obvious: Break open the closed path in the free energy expression and remove the zero mode factor 1/ T which accounts for all possible starting points. The path integral representation of eq. (5), analogous to eq. (3), is by construction equivalent to the Klein-Gordon propagator, i.e.

f ( d"q 2a)j

~ v

{6)

In Polyakov's case, of course, one needs to establish the procedure which ends up with eq. ( 6 ) having started from 1

G(x,x')= JL~jexpk,-p

d z ~

,

(7)

0

where [ C J x ( z ) / ~ h ( r ) ] is the reparameterization invariant measure of functional integration for the particlelike excitation. One's basic task, then, is to show that the symbolic expression on the right-hand side has the actual meaning provided by eqs. ( 3 ) and (4), suitably adjusted to the open path situation if one is interested in the propagator. Once this is accomplished it is a trivial step to derive the correct expression for the propagator of a free scalar excitation. In accordance with our formal procedure, it turns out t h a t / t = too. Had we been more careful in our definition of parametric representations (2) and (5) we would have set lim~o f T in which case the mass parameter entering eq. ( 7 ) would have been identified as

dT/(T) ....

/~=mo + const./e. 404

(8)

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As already mentioned, our main effort in this note will be directed towards fermionic field theories. We shall proceed in two stages. First, we consider a fermionic field theory in d space-time dimensions which is coupled to an external set of SU (N) gauge fields (both d and N are arbitrary). The second stage is to incorporate gauge field dynamics. Our immediate task is no integrate over spinor fields. One could, of course, choose to proceed formally and perform the regularization on the resulting determinant [ 3 ]. We shall, instead, employ a regularized version of the fermionic field theory itself so that the whole procedure is well-defined from the beginning. Our regularized casting of the fermionic sector of the field theory reads as follows [ 2 ] : s R = i a d y.. ~ ( n a ) (TuD R - - i M R ) ~ ( n a ) ,

(9)

n,t.,

where

D~{u(na)=-

dayO,,f([yl)U(L . . . . . +v)~u(na+y)

(10)

and

M R ~ ( n a ) =m f ddyf( ] y [ ) U (L ...... +~.)~(na+y),

(ll)

d

with ~ , (z) d z , ) .

U(Lxx+v) --- P exp(i \

(12)

Lx,x+y

In the above formulae, f ( [y[ ) is a regulating function, a form factor which drops (very fast) to zero within a range comparable in size with the (space-time) cutoff a and Lx.~+ v is a line path from x to x + y in ~d which we assume, for simplicity, to be topologically trivial with respect to its embedding. We have also denoted .~,(x) gA ~ (x) T~, where g is the gauge group coupling parameter and Ti~ are the group generators in the R-representation. Finally, P in eq. (12) denotes path ordering. It is quite easy to see [ 2] that, in a continuous version of eq. (9), the limit in which the regulating function f ( [ y ] ) goes to a delta-function produces the customary action for an unregulated spinorial field theory with external gauge fields. Our regularization strategy amounts to defining the (covariant) derivative at the location na in a way which entails appropriately weighted contributions from a small, surrounding (continuous) spacetime region. Note that our discretization is performed within a continuous space-time setting and its sole purpose is to facilitate the definition of the integration measure. Our underlying logic is not to prevent a priori higher order (irrelevant) operators from controlling the passage to the continuum limit. At the same time we have erased, through the form factor, their impact over large scales. Let us see how our regularization strategy works. To begin, no ambiguity is involved in the calculation of the partition function. The Berezin [ 4 ] integration measure over Grassmann variables yields

Going over to the free energy we employ, being more careful this time, the parametric representation T R) e x p ( T M R ) - c o n s t . , l n Z ~ = T r l n ( i T . D R + M R) c~oTr i d -~-exp(iTy.D

(14)

c

which results from the behavior of the exponential integral Ei ( - c x ) in the limit c -* 0. As will become obvious shortly, c serves as an overall ultraviolet protection which will enable us to consider the limit a ~ 0 first. 405

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30 January 1992

F o r m i n g the trace in the representation p r o v i d e d by our discrete d-dimensional configuration space, we write

lnZ~,,,= tr

~

lim N~oo

O~a(n°all+ie(7"DR--iMR)Nln°o~) ,

(15)

,u= 1 n ° = --~

c

where tr denotes trace with respect to y-matrix indices. We have, o f course, employed the defining relation for the exponential operator

exp(iTy.DR+TM~)=

e=T/N.

lim [ I + i e ( y ' D R - - i M R ) N ] ,

(16)

N~co

Insertion o f complete sets of discrete configuration states per each " t i m e " step ~ leads to the result [2]

=

In Z ~ , , = tr

dr

- - lim

•

T

N .o \i=0

a,,o .....

e(?.pi+im)f(p'20~ 2)

X exp

i

y~ o~ } ~ i~= I n~L

exp - i

k/=l

d P'

~ j

exp

(

- i y=

ole--.:c~(nio~)

i=j

(-

,

) (17)

/=1

where f(pi20~2) is the Fourier transform ~ o f f ( [Yl ) and where .~,(n~
U(L ......... ),~ exp[--iol(ni--ni-I).~¢(n'oL) ] .

(18)

Note that the U-factors in eq. ( 17 ) close on loops e m b e d d e d in ~d, i.e, they form Wilson loop variables. It becomes quite obvious now that c represents the length o f the smallest closed path entering the s u m m a t i o n over T. It serves as an overall ultraviolet m o m e n t u m c u t o f f a n d this allows us to consider the limit o~--+0first. In this limit,f(e~Zp ~2) -+ 1 for all i. We may, therefore, write In Z ,,+o = tr

i

~d T -

~" 2/"

~x(r) ~p(z)

(';

dzp(r).2(r)

exp - i

0

)

1"

xP exp( f dr y.p(r)+iTm) P exp(-i f drk(r)..z/[x(r) ]) , 0

(19)

0

where the functional integrations over 9x(r) and ~ p ( r ) are defined by eq. ( 17 ). We are now ready to extend our considerations to the gauge field sector. Let us denote averages over gauge field configurations by ( ) ~. We shall not concern ourselves with the particular choice o f i n v a r i a n t integration measure over the gauge fields since our attention is focused on the particle-like excitations associated with the fermionic degrees of freedom. We should mention, on the other hand, our partiality towards compact formulations of gauge systems (see e.g. ref. [ 5 ] ) which a d o p t the H a a r measure o f integration and which is in step with the emergence o f the loop variable P exp [ - i~A~(z)dz~] in eq. ( 19 ). Consider now the partition function ~ for the full theory. It can be given as ~ = ( Z[,~¢] ).,+,

(20)

where Z[,~/] is the partition function inherited from the fermionic sector, i.e. the one entering eq. (19). We have inserted the ,~/-dependence o f Z for emphasis. If we ignore violent fluctuations in the gauge field, then we may write ~ We have extended the range of each p'-integration beyond the first Brillouin zone, the difference being a constant which vanishes as o~ - + 0 .

406

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PHYSICS LETTERS B

{ Z [ d ] )~e= ( e x p ( - F [ d ] )

30 January 1992

>~e~ e x p ( - (Flute])
(21)

and the free energy ~ for the full theory becomes f*=(F[.~])
lim ,.~o

i

dT T -

c

f

('f

9 x ( r ) G p ( r ) exp - i

x(O)=x(T)

drp(O'2(r) 0

)

7I"

×Pexp(f d'~[7.p(r)+im])tPexp(i~s_C,(z)dz,))~¢ , 0

(22)

CT

where CT denotes a closed contour in ~d of"length" T. The above formula constitutes our main result. It presents the free energy in terms of a functional integral over an assortment of paths which characterize a point-like entity. Moreover, the effects of its interaction with the gauge fields are embodied in the factor ( P exp [i fc~aCu(x) × dzu] ).,e. The latter is simply the expectation of a Wilson loop operator in a pure gauge theory. As such, it should depend on some general geometrical properties of the closed loop. To gain a more concrete insight on our result let us consider the propagator G(x, x'). According to what we have said before, the representation o f the propagator in terms of excitation degrees of freedom is

G(x,x')=

lim c~0

; s dT

c

('i

2 x ( r ) ~ p ( r ) exp - i

x(x) =.v x ( T ) = v'

drp(r).2(r) 0

) (; Pexp

dr [ 7 . p ( r ) + i m ]

0

)

q~T(Lx..~,) ,

(23) where we have set

&(L~,,-,)-/exp(-i

f ,~,(z)dz.))~s

(24)

Lx,x'

with L,.,x, representing a path in ~:~dfrom x to X' of length T. As with the closed path case let us assume that 0r(Lx.,,) depends exclusively on general geometrical characteristics of the path L,-,x,. We shall then carry out the functional integration without paying attention to this factor. The first exponential becomes, by partial integration, T

7

exp(-ifdrp(r).k(r))=exp{-i[p(T).x'-p(O).x]}exp(ifx(O.#(r)). 0

(25)

0

The integration with respect to 2 x ( r ) can now be performed, yielding the functional delta-function ~(/)(r) ). This means that the m o m e n t u m remains constant along each path (as expected). It follows that oo

G(x, x ' ) =

!!m° f d T

ddp

~exPl-ip.

,

(x'-x) ]

exp[T(7.p+im)]0T(L,-,x')

,

(26)

c

where p [ = p (T) = p (0) ] is the remaining unintegrated m o m e n t u m variable ~2 If the factor 0T(Lx.,-,) was missing, i.e. if we were dealing with a free fermionic theory, then T integration would give in the limit c--+0 the expected result

~2 Proper handling of the functional integrations proceeds of course, through our regularized expressions which define them in the first place.

407

Volume 275, number 3,4

PHYSICSLETTERSB

ddpd exp[--ip.(x--x')] Grree(x'X')=-f (27~) 7"p+im

30 January 1992

(27)

From the excitation point of view, the above result would have necessitated [ 1 ] the introduction of a spin factor 0(Lx.,,). One writes the following generic expression for the spin -1 propagator

Gr"~e(x,x')= y~ exp[iS(L~,.~,)]O(L,..~,,) where S(L,.~,)is proportional to the length of the path.

(28)

Lx,x'

The above symbolic sum translates into an integral over dT (path length) along with functional integrals over particle paths. The role of the spin factor is, precisely, to weigh these paths according to invariant geometrical properties. Viewed in this light, 0T(L,-.~,,) entering eq. (23) represents effects of the interaction on the excitation's spin factor. Consider, in particular, the situation where scalar fields are coupled to a set of gauge fields. Similar considerations with the spinorial case will lead to the introduction of 0T(L.~,x,) in the excitation path integral. Now, whereas the factor is unity in this case, the presence of 0T(Lx,x,) might affect the spin content of the particles in the theory. Such a situation actually occurs [6] in a ( 2 + 1 )-dimensional CP 1 model with a Chern-Simons term. Our analysis has given further support to the idea that geometry has a considerable role to play in field theory. Careful handling of integrations over field variables produces a remainder with a potentially rich geometrical structure associated with the excitation degrees of freedom. We consider it especially significant that we have been able, in this paper, to carry such considerations on spinorial field theories, given the stubborness of the latter to accept a non-perturbative treatment on the lattice consistent with expected continuum properties [ 7 ]. Our final remark pertains to our belief that there is a close parallel between non-trivial geometric properties of paths on the one hand, and non-trivial field topology on the other. Some evidence of this occurrence has already been given by the present authors [8 ] in connection with a compact casting of pure gauge systems. Let us conclude by displaying the relation between field topology and path geometry for a two-dimensional spinorial system. Consider the following gauge invariant action

q)(Lx,x,)

S=i~d2xd2y~(x) exp(i f A,,(z) dz~)~(x+y)7"Ouf(ly[),

(29,

X

where f ( lY] ) is a function &fast decrease. The above action can be recast into the form x+ V

S=ifd2xd2yf(lyl)lp(x)7~exp(i ~A~,(z) dzl~)[iA~(x+y)+O~v']ql(x+y).

(30,

x

If the path form x to x + y has no self-intersection the limit f ( [Yl ) -~ 32 (y) can be effected without any obstruction yielding the standard Dirac action f dZx g?(x)i0~(X). Suppose, now, the path from x to have a self-intersection occurring at point z. We can write (in obvious notation)

x+y does

U(L~.x+,.)=U(Lx..~+:)exp(i~A~(w)dwu)U(L,.+~,x+v),

(31)

Czz

Lx+=._,-+y

where L,. ,.+. and are open paths without self-intersections and C,~ is a simple closed path beginning and ending at z. It follows that the approach to the local limit must be accompanied by the condition

~A~,(w)dwu=2~q, 408

q=integer,

(32)

Volume 275, number 3,4

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30 January 1992

since only t h e n will the l o o p p o r t i o n o f the p a t h be c o n s i s t e n t with the D i r a c action as f ( l Y l )-~d(2)(Y). We have, therefore, a s s o c i a t e d t w o - d i m e n s i o n a l a b e l i a n i n s t a n t o n s w i t h a self-intersecting p a t h e m b e d d e d in two dimensions.

References [ 1 ] A.M. Polyakov, Gauge fields and strings (Harwood Academic, New York, 1987); Les Houches, unedited lecture notes (1988). [2] A.I. Karanikas and C.N. Ktorides, University of Athens preprint UA/NPPS-4/91 ( 1991 ), J. Mod. Phys. A, to be published. [3] A.N. Redlich, Phys. Rev. Lett. 52 (1984) 18. [4] F.A. Berezin, The method of second quantization (Academic Press, New York, 1966). [5] A.I. Karanikas and C.N. Ktorides, Ann. Phys. (NY) 199 (1990) 1. [6] A.M. Polyakov, Mod. Phys. Lett. A 3 (1988) 325. [ 7 ] K.G. Wilson, in: New phenomena in subnuclear physics, ed. A. Zichichi (Plenum, New York, 1977 ). [8] A.I. Karanikas and C.N. Ktorides, Phys. Lett. B 235 (1990) 83, 90.

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