A variational method for non-linear spinor theory

Nuclear Physics 44 (1963) 195-204; @ North-Holland Pub&h&g Co., Amsterdam Not to be reproduced by photoprint or micmfilm without written permissio...

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Nuclear Physics 44 (1963) 195-204;

@ North-Holland

Pub&h&g

Co., Amsterdam

Not to be reproduced by photoprint or micmfilm without written permission from the publisher

A VARIATIONAL METHOD FOR NON-LINEAR SPINOR THEORY KATSUHIKO SEKINE t Max-Planck-Institut ftir Physik und Astrophysik, Miinchen Received 6 December

1962

Abstract:

The generalized free field of Greenberg and Dell’Antonio is extended to the case when the Hilbert space has an indefinite metric, and is used as a trial function for calculating the ground state in the non-linear spinor theory of Heisenberg. The condition for the non-canonical quantization is derived as a c@equence of the variational principle.

1. Generalized Free Fields Generalized free fields were introduced by Greenberg I) and also by Dell’Antonio 2, as a generalization of the free field concept in which the commutator is a c-number but does not satisfy a homogeneous (linear) field equation. In the case of a spinor Ii/(x), the generalized free field can be defined by J/(x) = /d4p$(p)e-‘“‘”

(1)

with $(P> = %Jj-omd/O$,‘L

P+(&

p+Q2

+ e( - PJ

s

- 0 O~dlr(i)bt(,/L

p+(&

P@(P~ -0,

where the creation and annihilation operators satisfy the commutation

(2)

relations

(“(Ki 9 Pr), a+(“j 3 4s)) = IbCKi 9 Pr), b+(Kjs 4s)) = E(Ki)Btcix~6rs6(P-4),

C3)

and u(lci, pr) and U(IC~, pr) are normalized eigensolutions for particles and antiparticles, respectively, with mass xi. To the set of points {Ki> is attached a positive measure p(C) with finite weight. If dp(C) = ; cia(C-K?)dC,

(4

i=l

Ii/(x)

is a superposition of N independent free fields ti(4 = 4WI*,=O =,$Ci

$yfd’p

T {ai(pr)u(x,, pr)efP’“+b:(pr)u(~i,

+ Present address: Institut Henri Poincar6, Paris. 196

pr)eSip’“}

(5)

196

K. SEKINE

with

a,(pr) =- po~a(~/(, pr)l~=~,~ , etc.

(6)

{a,(pr), a+(qs)} = ( bi(pr), b f (qs)} = 8i~ij6rs6(p--q).

(7)

Then eq. (3) reads

It should be noticed here that the original definition by Greenberg and Dell'Antonio is extended to cases when the Hilbert space has indefinite metric; the parameter e i may be' positive, negative or zero. Then we easily verify that

= E ~,[c,12sl;)(x -x', ~,), i

= E ~,lc, I2 s ~( + ) ( x - x , ~

(8)

Ki),

i

where • (x-x')

,

(9)

and 10) denotes the "bare" vacuum defined by

ai(pr)[0>

-- bj(qs)[O> = O.

(10)

From eqs. (8) it follows immediately that

{~,~(x), ~(x')} = z - '(~,)~ ~(x - x'),

(11)

z - ' = 2 ~,[cil 2.

(12)

with i

2. Axioms of Local Field Theory

Greenberg has proved that the generalized free field satisfies the following axioms of local field theory: (i) relativistic invariance, (ii) the spectrum condition, stating that the energy always be non-negative and mass real, (iii) local commutation relations (micro-causality). The Wightman condition p > 0 (positive definiteness of the Lehmann weight) is, however, not satisfied by our extended definition of the generalized free field; the metric of Hilbert space is indefinite. In the next section, the generalized free field will be used as a trial function for calculating the ground state of the non-linear spinor theory of Heisenberg 3). It is

NON-LINEAR SPINOR THEORY

197

to be noticed that the axioms enumerated above, to be satisfied by variational solutions, are essentially the same as the requirements of Mitter 4). He studied the propagation function for the spinor theory, but by a different method of approximation. 3. A Variational Method for the Spinor Theory We start with the Hamiltonian

{½f d3x(~;(x)ek~O(x)~xk~XXk),k~(X))

~=z

-T-½12f d 3xxd3x2d3xad 3x4 K(xl x2; xsx,)(~}(x,)O~k(x2))(~}(xa)O¢(x,))},

(13)

and the commutation relation for equal times defined by (11). The factor Z is left undetermined. We have introduced a form factor K(xlx2; x3x4) representing a space-like smearing out of the interaction. A suitable local limit will be defined later. A state vector can in general be expressed as i~ )

= rn=0

25

d 4 x l . . . d4xm

;

d4yl . . . d4y.

x A,,n(xl... x,,[yl.., yn):~(xl)... O(x,,)i~(yO...i~(yn):]0),

(14)

Am, being the probability amplitude for the specified state. The variational principle is now

<~1~i~> = EUAo~;~,, c,]

= extremum

(15)

under the normalization condition (4~14~> = 1. To evaluate the matrix element ( ~ l ~ f ] ~ > it is useful to transform the Hamiltonian into the following form. From the definition of the normal product 5), we have

X=z{~fdSx:(~(x),k°°(x) eff(x),kO(x)): 8Xk

gXk

+½fd~x

e[a>

= _26(o)z Z ,lc,12 f

(1+

]

T- 2(~-~7036(0)~ ezlcii2~ eACg[2 dSpdSq~.(p.q; p-q) x¼ Tr I757~ (?4+ i~' " --P-K'I

(74

!~" q-K~]l

(23)

NON-LINEAR

SP1NOR TItEORY

199

where K is the Fourier transform of the form factor /.

K(xlx2; x3x,,) = (2~)9j|dakldak2dakaK(kl X

k2; ka) e ikt

"(xl

- x 2 ) +ik2 " (xa-x,l)+ik3 " (x2--x4)

(24)

The factor 6(0) means the value of &(k) at k = 0. If the system is enclosed in a cubic box of finite volume V (with periodic boundary conditions), we have the factor V in place of (2n)36(0), as has been established by Hugenholtz 6). He has shown among other things that (1) if a diagram without external lines is called "vacuum diagram", each isolated connected vacuum diagram gives a contribution proportional to V; (2) the'vacuum energy is thus proportional to V as far as only connected diagrams are taken into account. Specializing the form factor as 11 only for Ipl, lql < a K(p, q ; p - q) = ~0 otherwise,

(25)

we have finally

E V

f--

:z z

(2~) a i

(, + p2 + x~l

12Z + (2n)6 ~ eilCilz ~ eJlC'lzfadspf adSq (1 + x / ~2x~xj ~

,1 "

(26)

Variation with respect to Kj gives d3p

~, (3f'4p2+4 = + ~-~-3 2

x]f 4

d3P

(p2g-;~),/ 4g+,,t/~; 4q~+,,,

~,1C,1%

2

r ~ f A (q2 d3q + , O ~ l~ " (27)

For definiteness we have introduced the upper limit A -~ oo of the integrals also on the left hand side. Summing over j we have the equation 2/2

-

,t .

4q~

+~

(q2+,0~I,

(28)

where

f

.4

dap

E.,~Ic~I~I (P~+~)~ cA =

[.~

'

~AcA%J

d3p

,/g+4

(29)

'

200

K. SEKINE

assuming [lima~o cal < co; otherwise the equation would be meaningless. If we put

p( 2) =

.,ic,12a( 2_ 4).

(30)

( dK2p(/g2)x2 ( ( fooodx2p(x2)f Ax / qdaq a (qZd3q 2 + ~ 2 Jo +-~2)~] "

(31)

J

then 2/2 3+ca = + -- ~ - ~

Fromthis relation we can see that fo°d~:2p(x 2) ~ o~d~2p(/~2)g 2

~

aA-2,

(32a)

b(log A)-*

(32b)

The only assumption made is that the integrals in question exist. Eqs. (32) imply that

fo°dx2p() K2

=

0,

fo~d,~2p(/¢2)~C2

=

0

( ) 33

in the local limit A --+ co. This is just the condition that the commutation function is free from the 3 as well as the ~' singularity on the light cone (non-canonical quantization7). Strictly, the p function in the above is not identical to the usual spectral function because in eqs. (8) the symbol [0) means the "bare" vacuum, not the true vacuum. But in the approximation made here, that is, under the assumption (22), the difference mentioned can be neglected. Determination of the spectral function in more detail and of the explicit form for the extremum energy has not been carried out. It is, however, possible in principle (see appendix). It should be remarked here that the variational calculation gives, in the usual case of positive definite metric, an upper bound of the eigenvalue, whereas this is no longer true in the case of indefinite metric. The value calculated from the variational principle will approximate the true one, but it may be greater or smaller.

4. Spectrum Condition. Translation Invariance o f the Vacuum

Pu

Let us define the energy momentum operator as the infinitesimal generator of the translation operators. According to Haag and Schroer 8) the spectrum condition can be formulated in two ways: (a) The four-vector formed by the simultaneous eigenvalues of the four components of Pu never lies outside the forward light cone (i.e. the energy is always non-negative and the mass is real). (b) There is a state in Hilbert space, called vacuum, which has zero energy momentum. The requirement (b) is stronger. Since we are now interested exclusively in the ground state, let us consider the spectrum condition in the sense of (b). We postulate

201

NON-LINEAR SPINOR THEORY

the existence of an invariant vacuum (invariance under translations) =

(34)

0

or, at least, the vanishing of the expectation values

(35)

< a l P . l o > = o.

Now, the space components Pk are given by Pk =

-

dsx

O (x)

(36)

OXk

which are equal to those corresponding to the theory without interaction. This is due to the fact that the non-linear term does not contain any derivative of the field operator. Taking into account the definition of the creation and annihilation oparators, we see immediately that Pkl~> = 0.

(37)

On the other hand, for the fourth component of Pu, that is, for the Hamiltonian, we postulate (38)

< l elo> = 0.

The left hand side of this equation is given by eq. (26). Therefore,

12f

f

fA~A d3q (1 + ~¢/pZ+ ~c~x/p-~+~c-;i]' 2KK' __~

= --- (27@ d~c2P(xZ) d~:'ZP(X'2) dap For sufficient large values of A, we have 2 f dx2p(~2)(a 2 + x2) = ___ (27z)3

dKZp(xZ)A2

+2

dtc2p(~c2)tca

.

(39)

As can be seen, this equation is compatible with the conditions (32). Thus, the postulate (38) is compatible with non-canonical quantization, only giving some further information on the behaviour of the p function. It should be noticed that eq. (35) has a Lorentz invariant meaning; all observers find vanishing energy momentum in the state If2>. 5. Conclusion

By using the generalized free field as a trial function, we have derived in the nonlinear spinor theory the condition for non-canonical quantization as a consequence

202

K. SEKINE

of the variational principle. It is to be noted that the generalized free field satisfies Lorentz invariance, the spectrum condition and micro-causality. On the other hand, we have shown that the generalized free field with the Lehmann weight p is, though approximate in the sense of the variational method, a solution of the non-linear spinor equation only if the function p satisfies the condition of non-canonical quantization. As is immediately seen, no positive definite measure satisfies this condition. This implies the existence of an indefinite metric in Hilbert space. Thus we have proved, albeit in the framework of a variational approximation, that the indefinite metric is a mathematical consequence of the non-linear spinor equation. Mitter 4) earlier attacked the same problems but by a different method of approximation. By means of the new Tamm-Dancoff method he derived a non-linear equation for the SF function. But his solution satisfies only part of the conditions, namely only the first equation (33), for non-canonical quantization. The author wishes to express his sincere thanks to Professor W. Heisenberg for his kind interest and illuminating discussions throughout this work, and particularly for the hospitality extended to him at the Max Planck Institute. Thanks are also due to Mr. J. A. Swieca for his helpful discussions. The support of the Alexander ~¢on Humboldt Stiftung is gratefully acknowledged.

Appendix The same variational principle can be applied to a simplified model: the anharmonic oscillator. Consider the system defined by ~ o = ½p2+¼2q,,

(A.1)

[p, q] = -- iC,

(A.2)

with where C may be positive, negative or zero. Now we put

p= o~(-i)~2 (a-a+)+fl(-i)V2 (b-b+ ), (A.3)

1 (a+a+)+fl

1 (b+b+) '

with [a,a +] = 1,

[b,b +] = - 1 .

(A.4)

We have thus introduced two kinds o f " b a r e " particle operators obeying the commutation relations for positive and negative metric. The corresponding masses are x and #. The weights ct and fl are related by ~2 (~t and fl are assumed to be real).

f12 = C

(A.5)

NON-LINEAR$PINORTHEORY

203

A state vector can in general be expressed as

'y' c,.. (a +)m (b +)n [0>,

1+> --

(A.6)

.:o.:o

the "bare" vacuum 10> being defined by (A.7)

a[0> = b[0> = 0.

The energy (A.8) is now to be made an extremum under the normalization condition. We try the simplest "ansatz" for the ground state:

CoolO>, Icool2= 1.

(A.9)

Then the eigenvalue equation is ----

-E

Coo = 0,

(A.10)

from which we have (A.11) By differentiation

OEoK__ = ¼°t2-- 2(3J++ ~- 8 /~2 ¢ O#

-- ~ ) - - - 0, (A.12)

8# z

Hence x = . = ~/}2C,

(A.13)

with C = 0t2-fl2. Therefore, the masses of the normal and the ghost particles are equal. Substituting (A.13)into (A.11)we have E = ~ ~/~-~C~.

(A.14)

Thus the extremum energy depends only on C; the ratio of mixture st/flremains undcterrnined. The above result can easily be generalized to the case when more than two kinds of normal and ghost particles are present:

P = E °ti(--i)x/-~(a,-- a;'), i

q = ~ e i ~ ( a i +1 a i ) , i ~/2xi

+

(A.I5)

204

K. SEKINE

where

[al, a +] = el, Z

gi = +1,

(A.16) (A.17)

= c.

i

F o r the same trial function (A.9) we have n o w ,

/g

.

(A.lS)

Hence

OE

<

k~i Zz_ 3~ i~2 (~i giOt~ 1 )

s-7

7,"

(A.19)

We can easily see that all xl are equal: rq = ~/-}2C.

(A.20)

The extremum energy can be expressed again in the f o r m (A.14), where C is now given by (A.17). The weights ~i remain undetermined. This reduction m a y be understood as follows. The a n h a r m o n i c oscillator is described by the Hamiltonian (A. l) and the c o m m u t a t i o n relation (A.2). The representation o f the c o m m u t a t o r ring is uniquely determined (up to unitary equivalence) so long as the n u m b e r o f degrees o f freedom is finite 9). That is to say, the Hilbert space for the anharmonic oscillator is as large as that for only one harmonic oscillator with a single value o f frequency. I f we take as unperturbed system a n u m b e r o f oscillators with different frequencies, the corresponding Hilbert space is too large to represent the states o f the given (perturbed) system. It is, however, expected that this is not the case in actual fidd theories because o f the infinite n u m b e r o f degrees o f freedom. As is known, there is an infinite n u m b e r o f different representations for one and the same c o m m u t a t o r ring so). It is quite likely that the existence of a non-trivial mass spectrum is connected with the situation mentioned. This problem will be discussed in a forthcoming paper.

1) 2) 3) 4) 5) 6) 7) 8) 9) 10)

References O. W. Greenberg, Ann. of Phys. 16 (1961) 158 G. F. Dell'Antonio, J. Math. Phys. 2 (1961) 759 H.-P. Dtirr. W. Heisenberg, H. Mitter, S. Schlieder and K. Yamazaki, Z. Naturf. 14a (1959) 441 H. Mitter, Z. Naturf. 15a (1960) 753 G. C. Wick, Phys. Rev. 80 (1950) 268 N. M. Hugenholtz, Physica 23 (1957) 481; W. R. Frazer and L. Van Hove, Physica 24 (1958) 137 K. Sekine, Nuclear Physics 23 (1961) 245, Cahier de Phys. 143 (19~) 261 R. Haag and B. Schroer, J. Math. Phys. 3 (1962) 248 J. von Neumann, Math. Ann. 104 (1931) 570 K. O. Friedrichs, Mathematical aspects of quantum theory of fields (Interscience Publishers, London, 1953); L. Van Hove, Physica 18 (1952) 145; R. Haag, Mat. Fys. Medd. Dan. Vid. Selsk. 29, No. 12 (1955)

A variational method for non-linear spinor theory

A variational method for non-linear spinor theory

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