A method of regularization in field theory

8.C~ ......_.___....._._At

Nuclear Physics 40 (1963) 309--320; (~) North-Holland Publishing Co., Amsterdam Not to be reproduced by photoprint or microfilm without written permission from the publisher

A M E T H O D OF R E G U L A R I Z A T I O N I N F I E L D T H E O R Y M. A. MARKOV P. N. Lebedev Physical Institute, Academy of Sciences, Moscow, USSR

Received 11 July 1962 Abstract: Functions transforming into plane waves at t --> -t- oo are used instead of plane waves for the construction of the S-matrix. The "free field" quantum acquires the meaning of a particle

only at t--->:t:~. The formalism is constructed in this way: a certain"counterfield" rf(x) damping in time is added to the usual free field cp°(x). This counterfield, quantized according to indefinite metrics ("with the reverse sign"), regularizes the AO-functions (sects. 3,4). The quanta of the "counterfield" q0'(x) disappear at t -+ ± oo (sects. 2,7). The "counterfield" makes no contribution to the probabilities and cross sections for the observed effects (sect. 8). The theory is causal (sect. 5), unitary (sect. 6) and relativistically invariant (sect. 8).

1. Introduction

Despite the formal and inconsistent character of the Pauli-Villars regularization this procedure is widely used in field theory 1). The probable explanation is that accidentally or otherwise (from the viewpoint of the future consistent theory) this procedure imparts the meaning of finite expressions to the divergent integrals of the current theory, without contradicting either causality or unitarity. However, the problem of making such a procedure theoretically consistent seems attractive enough since the resulting theory would automatically include the regularization in question. The Pauli-Villars regularization is essentially the formal introduction, at a late stage o f the procedure, of certain "counterfields" changing the f o r m of the propagators, but leaving intact (and this is essential) the state vectors. A question arises: to what extent will the principles have to be reformulated to obtain automatically the regularization under discussion? It is a fact that real counterfields regularizing the propagators can readily be introduced. But the difficulties to which they give rise have not been surmounted so far 2). Propagator-regularizing counter fields are also introduced in this paper. But they d a m p in time and there are no quanta of these counter fields either in the initial or final state of the system (at t -~ + o0). The division of the field into "field" and "counter field" is artificial: it is only a method convenient for the comparison with the present conventional theory. The physical essence of the present theory consists in that the field quantum is interpreted as a free particle only when t -~ + o0. In the conventional theory the creation of a particle is regarded (disregarding the uncertainty principle limitation A t A E ~ h) as the instantaneous appearance of a free 309

310

M . A . MARKOV

particle with a certain rest mass. In the scheme under study, during the short lifetime of the particle created its mass is uncertain, as well as its energy. The components distorting the conventional plane wave damp in time and the state assumes the meaning of a free particle state, as interpreted in the conventional theory. The scheme is symmetric with respect to the production and absorption of particles, as well as with respect to time.

2.

Counterfield

As an example let us first consider the scalar field. To each plane wave, describing in the conventional theory a particle with momentum k and energy ~/k2-Y+--M~, e +ilk" x - ( k 2+

MZ)X~t]

(2~/~ + M2).,t"

(1)

is attached a certain wave "satellite" or "counterfield" characterized by the same value of the vector k, but represented by a certain integral over the mass parameter m:

e+_ik.x ~T't(k2+rn2)½t p

|

,.

dm

~

Whereas the usual field quantum is connected with the plane wave (1), the "eounterfield" quantum is connected with a more complicated concept of the type (2). It is by functions of the type (2) that the counterfield is quantized here. In other words, the corresponding operators of creation and annihilation of counter field quanta are multiplied by functions of the type (2). The conventional free field (p°(x) used to construct the S-matrix is written in the form

1 l" dk e~ik%~(k)

(3) = k. o(x ) =

'

(4)

where + and - denote positive and negative frequency functions. We proceed from the following explicit form of the counterfield operator functions: I

H, ( / ) ,

(4)

dk dm {p'-(x)--(~)~J-~ooe-ekXTq}.(k)f(--~) V~-H.(1).

(5)

=

1

~

f

_

m z

--

A METHOD OF REGULARIZATION IN FIELD THEORY

f(m2ll2)

311

Here is a function which decreases rapidly as rn increases; the form a complete set of orthogonal normalized functions such that

~H.(1) Hn(m---/)=

H,(m/l)

6 (m rn-----~'), /

(6)

where l is a characteristic constant of the theory. The Hermitian functions t are here chosen for the functions H~, but Bessel functions or any other set of functions could be treated in the same manner. Eq. (2) describing a quantum of an attached field in the mass state n titus assumes the explicit form

and a sim.lar expression for to2k • The integration limits are still uncertain, but in a particular case one may take S_+~. It is essential that functions damping in time be chosen. Owing to the oscillatory dependence on time of the integrand (7) the damping in time of the to~ functions is characteristic of a wide class of functions In particular, selecting in eq. (7) H~ = Ho oc e -"~m2, and f = 1, we obtain in the rest system (k = 0)

f(m2[12).

+

to.,k=o

f+°

e_imt_m2]212

-at

dm ~

e_~lat2

(7')

l

i.e., to0,1t,= + o actually damps in time and the damping can be intensified by a suitable choice of the function The damping of the function (7') is symmetric in time. A privileged instant is selected in the function (7'): it corresponds to the time of the creation or absorption of the particle (t -- 0). A counter field arises only in interacting systems. The " t r u l y " free field (t -* + oo) coincides with the field of the conventional theory. The field t0°(x) satisfies the usual free field equations. The counterfield is defined as an accompanying field, according to eqs. (4) and (5). The general field is not supposed to satisfy any equation.

f(m/l).

3. Quantization The functions ~°(x) are quantized in the usual way. The corresponding quantum brackets for the eounterfield to' are taken with the "opposite sign": [to;(k); to~ (k')] = - ~ ( k - k')~',,

[to2(k); to~,(k')] = l-to:(k); to.+,(k')] = o,

(8)

[to°(x); to'(y)] = o. * M o r e p r e c i s e l y , H e r m i t i a n p o l y n o m i a l s w i t h a n o r m a l i z i n g f a c t o r , e.g. H 0 =

1 • e-mU2

t2.

31~

M . A . MARKOV

In virtue of eqs. (8) we have E E

d(~'Y-'~,

1 a

7

ef(kmy-krn'X) m2

mr2

2~/kmkm,

"

g dm dm' l

l

(9)

In virtue of eq. (6) we have [~p (x),~p

(y)] = -

2k o

I

1

=-,f+_]~-A+(y-x,m~)f(~)~-=-A'+(y-~)

(9 ')

(10)

if we choose

f fm@) am' - 7 - -~1 . All other

A '( )-functions

(11)

of the counterfield have the same structure:

--A'() ~ +

-ooA ( ) ( x , m 2 ) f 2 -~-

l

l

(12)

Using the properties o f the operators ~o~ (k),

(tp;(k)9+(k'))o = 5(k-k'), (tp+(k)q~+(k))o =

O,

(¢p~-(k)q~-(k')) o = O, (¢p°±(k)~o'n-+(k))o = O,

(13)

it can be shown that

(T(~o'(x)¢(y)))o=l(~°'(x)q~i(Y))° t(~o'(y)~ (x)> o

= _I A,(_ )

for x ° > yO

i

= iA '(+)

for x ° < yO,

and d 'c=

i(Tq)'(x)tp'(y))o.

(14)

313

A /vIETHOD OF REGULARIZATION IN FIELD THEORY

Here we perform the usual extrapolation of eq. (14) for the case x ° = yO. Thus, with the aid of the functions H,(m/l) with the properties (6), the field q¢ can be quantized with respect to the time-damping functions of the type (2) or (7).

4. Singularities of Propagators The propagators of the field q~O+q~, are differences of the corresponding propagators. Thus

(15)

= 4,-a".

In the neighbourhood of the light cone we have

1 ~5(s)+- - 1 .M z M 2 O(s) - - In ½ s ÷ M A~ ~, -~ 4~2is + t 8r~2 16sT c

(16)

and hence M2 Ar~, = i - - In 8z~2

½s¢M-

M2 " ~ f m2 In 1 6 ~ 0 ( s ) - 8rc2d

(½s÷m)f2

(m_~) [ml dm ~ / l l

+ 0(s)16~d f m2f2

(m_~)Imll dml

(17)

According to eq. (17), the main singularities of the A~-funetion (5(s) and I/s) are eliminated by the introduction of a counterfield, irrespective of the specific form of the fZ-function. Or rather the only requirement is that f 2 be a normalized function, i.e., satisfy the condition (11) or even the condition

f~f2 m(-~)lmldrn-1 l = 1.

(18)

To eliminate the logarithmic divergences in the A( )-functions and discontinuities at the points S = 0 the form of theffunctions has to be specified. The At) functions of the total field (A~g) can be written in the form

Z~ oz f A( )(s, m2)p(m2)dmz,

(19)

where p(m 2) = 5(m 2 _ M 2) _ f 2 Choosing

f2

.

(20)

in this form, for example = ½a2 Re

lm

,

(21)

~14

M . A . MARKOV

we obtain the A( )-func'ions we considered in ref. 3). They have singularities not on a cone but on a hyperboloid. The functionsf 2 of the type (21) alternate in sign, which leads to imaginary ~/f~ and consequently changes the Hermitian properties of the ¢#' field operators. Postporting the treatment of more complicated situations, we take up here the cases of real f functions. It should be noted that it is hardly worthwhile to strive for a "complete" regularization of the A function, i.e., for the equations Sm2"p(m2)dm 2 = 0 for any integer n, since the elimination of the most important singularities from the A() functions is sufficient for constructing a theory without divergent values of the observed quantities. Indeed, in the momentum representation Ar~g will be of the form (taking into account eq. (11))

j ~-2~)(-~_

~T-_/-~

(22)

Thus, when calculating the order of divergence of the relevant integrals each inner line of the Feynman graph in the theory here developed contributes to the denominator of the integral by two orders of the momentum (/,2) more than in the conventional theory. This means that all divergent integrals in any conventional variant of field theory (including the four-fermion interaction) become convergent in the theory under discussion. 5. Carnality The commutators of the operators q¢(x) and q~'(y) are written in the form A' =

m2)f2

[ml dm l l

(23)

It can easily be seen that if m does not assume imaginary values, or rather if m 2 is positive everywhere, A' vanishes outside the light cone, i.e., in the region = c 2 ( t - t') 2 -

< 0.

Indeed, the integrand function in eq. (23) can be written in the form x / ~ ~ e(xO)O(s)J~(mx/~), A ( x - y , m 2) -----~1 e(xO)&(s2)- 47rx/---1

O(s) = 0

(24)

s>0 s < 0.

Since O(s) does not depend on rn, we can see, after substituting eq. (24) into eq. (23),

A METHOD OF R E G U L A R I Z A T I O N I N F ~ L D THEORY

~l~

that the property of the conventional theory commutator to vanish for s < 0 holds in eq. (23) for any f a c t o r f 2 normalized by eq. (11) or (18). The only restriction is the condition rn2 > 0 for the validity of representing the A-function as in eq. (24). Thus, the condition for the localitity of the operators q~, [~0(x), ~o(y)] = 0,

(1)

ensures (if s < 0) the fulfilment of the causality principle by the S-matrix constructed with the field operators (sect. 6). 6. The S-Matrix For the sake of simplicity let us consider the interaction of two scalar fields. One of them is a field q~M,a quantum of which carries the mass M, while a quantum of the other scalar field ~Oo corresponds to a particle with zero rest mass. The field q~u consists of the field ~0° and counterfield ~p~ defined according to eqs. (1), (5) and (11):

~0~, = ~0° + ~0;,. To simplify the analysis we treat the field ~k° as a conventional field (without a counterfield) satisfying d'Alembert's equation. The field operators ~o~t and ~k° are Hermitian and local:

oot

~otu = ~ou,

[qu,(x),-q,*(y)] = o,

= ~,o;

[~,O(x), ~,O(y)] = o

outside the light cone. We write the interaction Lagrangian as ~ ( x ) = O~0,(x)~,°(x),

(25)

~t(x) = ~(x),

(26)

[ ~ ( x ) , .o~°(y)] = 0

(x-y space-like).

(27)

Just as in the conventional theory, the S-matrix is of the form S = 1 +.=~l=

T ( ~ ° ( x l ) . . . ~ q ( x . ) ) d x l . . . dx..

(28)

It can easily be checked that the unitarity condition written as the following relation l) holds:

s. xl

.+

1~.-1 -

-

x~;

_x

Sk(xl...x~)S'(x~+,...x.) =

0,

(29)

316

M.A. MARKOV

where ~ ( x l • • • Xk)/Xk + 1. . . X,) designates the sum over all n ! / k ! ( n - k ) ! divisions of the set of points x 1 . . . x, into two sets of k and n - k points. Here permutations inside each of these sets are equivalent owing to the symmetry of Sk with respect to the arguments (28). As in the conventional theory, the fulfilment of the condition (29) is a consequence of the Hermitian property (26) of the operators .oqPand the possibility of writing S, as a T-product (28) arises from the locality of the operators *~P which is a consequence of the locality of the operators cp and ~k.

7. Norm, Vacuum, Damping States In the scheme under study the amplitude of, say, one-particle states consists of two terms: o , (30) The term ~o is identical withthe states described by the conventional field theory 1): ~o = f c(k)~+(k)dk~ ~

= ~i ~o".

(31)

The term ~ n describes the state of the attached counterfield, with the same as yet unnormalized distribution over the momenta c ( k ) , but in the mass state n: ~i,,, =

f

c(k)q~*,, (k)eU"°~f ( ~ 22) H,

(-~) dm T ~oII ~oI = ~ , ~ .

(32)

A "second vacuum" ~ is here introduced asin the well-known Heisenberg treatment. The entire Hilbert space is thus divided into two spaces. One of them (states ~ ) coincides with the Hilbert space of the conventional theory. The second Hilbert space (states ~o~) is auxiliary. All excited states ~ damp in time and vanish at

t ---> + oo. The states of the first and second vacuum are normalized:

(~0*In%I ~'0 ~- 1~

'~*IIcBII = 1.

~0 ~0

The general vacuum ~o is described by the product of the functions ~

and ~ :

=

The norm of the state (30) under study, taking into account the properties of q~+(k) and q~ff(k), has the form ~g>1

*

=

f c * ( k ) c ( k ) d k - / rd ; d

x fc*(k)c(k)dke"'~"'°-k"'°)f(~)V~H"(7)ft~) /m'2\]/Im-~'l "l H,,(~) -

-

.

(33)

A METHOD OF REGULARIZATION IN FIELD THEORY

H.(m/l)

317

f(m2/l)

The functions and are chosen so that after the integration over m the states (32) decay rapidly in time. Thus,

~* ~1 ->f c*(k)c(k)dk

for t ~ + oo.

(34)

In other words, the normalization of the physical states when t -~ + oo coincides with the conventional procedure. We can arrive at the same conclusion in a more general case, without the restriction to a definite mass state n. The limiting case n=o lna~ with equal coet~cients a n admittedly calls for a more detailed study and somewhat different arguments in favour of the same normalization (34), involving the analysis of the probabilities for the respective transitions which, as appears from sect. 8, occur only between functions of the class (31).

8. Probabilities and Cross Sections An essential feature of the theory developed is that the counterfields m a k e no contribution to the probabilities and cross sections for the physical processes. Rather, their role reduces to the regularization of the AO-functions. In the conventional theory the squares of the corresponding matrix elements contain 6 Z ( E ) ~ and the probabilities defined as

Td(E)

(M) 2 to =

r~

(35)

T

do not depend on time. For the transitions considered, containing counterfields instead of 62(E), there occur expressions of the kind

f 6(E, m)Fn(m)dmf 6(E, m')F~(m')dm'

(36)

which do not increase with time, and the corresponding probabilities tend to zero: to' - ( M ) 2 r-,~ T

÷ 0.

(37)

It is assumed that the initial and final states of the counterfield can be realized in a certain arbitrary but unique * state n. Transitions between field and counterfield states should not occur according to the very meaning of the introduction of a satellite field, and they do not, as we have seen, in the formalism developed. The above treatment is valid for matrix elements of any order: the rule that the procedure reduces to the matrix elements of the conventional theory, but with regularized Ac functions, is valid in all cases, t Unfortunately, this problem is not actually so simple: this additional requirement which excludes the final states Z~_otP'x~anwith (a~ ----a,~)may lead to a violation of unitarity. This problem requires additional treatment.

818

M . A . MARKOV

9. Fermi Field For a Fermi field the counterfield ~'± is of the form ~'-+(x) = ( 1 ) ~ ~fdke+'~k~(m,

(38)

k)f (~2 ) H~ (1) dm_i_

where

~b+(m,k) - O(k°)~"(m"k),

~2(k, m) = O(ko)~(-k, m)

2ko

2ko

(ko = x / ~ ) . Let + ~p,~(k, m) = ~ a~(k)Y/'~+(k,m).

(39)

The counterfield ~b'±(x) is postulated in the explicit form of eq. (38) in which the operators a~(k) are independent of the parameter m. It is postulated that [a~(k), aj;,(k')] + = 5(k- k')t~ijS~,.

(40)

In contrast to the conventional theory, the sign is reversed here in the quantization of the counterfield. Let E "//'~-(k)~'~+(k) = ---1, 2

((~+(m+M)~ \

2k o

]

(41)

~#

where # is the mass of a particle. The corresponding anti-commutator is of the form

q/':'(k, m)q/';l'(k ', m)

[~k:-(x), ff~+(y)]+ = ( 1 ) 3 ~ ~ f dkdk'e'(k'Y-~x)~

× [a~n(k),at,n,(k )1 +f

f \ lZ ] Hn

Hn

V

l2

l

l

On the basis of eq. (40) we have I A

!

~zrt/

l~""

~"

L,

d

r

¢c

~"

~

d

v

V/-mllm'l S~ X ~

i2

y Hn

/

2

/

\ 1 /

(/)

-~-

( ~ ) d m dm' Hn

I

(42)

l

On the basis of eqs. (6) and (42) we have 1 3 +o~dm dke,k(,_.) 2 (m2~[~.+m+M~ [1+=--(2,) f-o~ l f ~f ~-~'~ ~ o " " # ~

(43)

A M E T H O D OF R E G U L A R I Z A T I O N I N I~IELD T H E O R Y

= _ (':]'f+OOdm:

\~/J-.

(~-f) L~-(i?~o~ +M) iA+(y-x,m)

l-

=_ (l~af+~dmf2 (~-~2) L~-{is+(y-x,m)+iMA+(y-x,m)}.

319

(45)

(46)

The equivalence of eqs. (45) and (46) is based on the fact that

f'5

o ,_-

owing to the parity of the integrand. When deriving the anticommutator in the form (46) we could, instead of introducing the somewhat unusual normalization (41), represent the counterfield ~'(x) as the sum of two fields: ~'(x) = ~ l '( x ) + ~ / M ~- , 2 , '

(47)

where instead of eq. (41) the normalization for the ~g" and "//'2 is

Z "¢/'l,(k)'/:lt~(k)=\-~-ol~,#' .

v=

I,

(48)

2

~//'2~(k)~/'2# (k) =

(49)

In the theory under study there is no time description of events. Nor does the construction of the S-matrix rest on equations of motion of the "free" field. In the conventional theory time, unlike the space coordinates, is not an operator, but a c-number: [elx]

¢ o,

[ e , t] = o.

(53)

This time-space asymmetry can be traced to the Dirac equation and the representation of the Lorentz group 4) it selects. In other words, in virtue of the equation of motion, energy is expressed through the momentum E z = p2 + rnz, which commutes with time. The result does not depend, as we have seen, on the choice of particular functions H n for the initial and final functions of the counterfield: this corresponds to the fact that in the framework of the present theory there is probably no possibifity for choosing particular Hn functions. The theory does not define concepts like the energy and momentum tensor or conservation laws in the form of divergences. That would be a far too detailed description for such a theory. However, when t ~ +_oo energy and momentum acquire the meaning of the respective quantities of the conventional theory, and the 6 functions arising in the matrix elements ensure the corresponding conservation laws.

320

M.A. MARKOV

The problems o f gauge invariance in electrodynamics are not considered. In general, gauge invariance is ensured by the regularization of the products of the Acfunctions a n d not by the functions Ac separately as in the case in the present theory. However, we have here an arbitrary functionf(m2/l) the m o m e n t a of which can be used. Generally speaking, there can hardly exist a consistent theory for a single field (electrodynamics, for example) aside f r o m other fields. Therefore a theory of the Heisenberg type appears to be a more consistent scheme. It is for further investigations, in particular of four-fermion interactions, that the above formalism is developed in this paper. The author is grateful to A. A. K o m a r for numerous stimulating discussions.

References

1) 2) 3) 4)

N. N. Bogolyubov and D. V. Shirkov, Introduction to Quantum Field Theory (Moscow 1957) A. PaLs and G. E. Uhlcnbeck, Phys. Roy. 79 (1950) 145 M. A. Markov, Nuclear Physics 10 (t959) 140 P. A. M. Dirac, Roy. Mod. Phys. 29 (1949) 392