SUPPLE~IENTO AL " ~ O L U g E III, DEL

SERIE X

N.

4, 1956

1o Semestr~

NUOVO CIMENTO

The Non-linear Field Theory and the Theory ot Relativity. D. I. B L 0 ~ C E V Academy o] Sciences of the U S S R - Moscow

CONTENTS. - - t Introduction. - 2. A classification of non-linear theories. 3. - Equations of class A. - 4. Equations of class B.

1.

-

Introduction.

In recent years, the development of the q u a n t u m field t h e o r y h~s been m~rked b y successful gpplic~tion of the m e t h o d of reuormalizing the m~ss o f the particles. However, this m e t h o d is only a m~thematic~l procedure t h a t mgkes it possible to circumvent phenomena add processes relating to v e r y high frequencies and to v e r y sm~ll scales. I n addition, this methQd is not. ~lwws applicable. F o r this reason, ~long with the development of methods of renorm~liz~tion, various theories were developed in which the divergence of the self-energy of the pgrticles is excluded not b y ~dditiongl methods b u t on the b~sis of t h e initial physicgl content of the theory. The following two trends gre represented b y gn especially large n u m b e r of works: 1) the non-local field theory, ~nd 2) the non-linear field t h e o r y . I n this report we shall consider the m~in peculiarities of the second t r e n d , the non-linear field theory. The introduction of a certain field scale % is characteristic of this t h e o r y ; thus, indirectly, there is introduced a certain length So = ~r (where g is the charge of the particle), length which m a y conditionally be regarded as <(the size of the p~rticle ~>. The following m~in problem is t h e n discussed: to w h a t e x t e n t is the nonlinear field t h e o r y capable of eliminating, in principle, the a b o v e - m e n t i o n e d difficulties of the q u a n t u m t h e o r y ? I t is not necessary to analyze the concrete f o r m of the t h e o r y in order to investigate this aspect of the problem. I t is sufficient to examine its general features.

630

2. -

D.I. BLOHINCEV

A Classification

of N o n - l i n e a r

Theories.

Non-linear equations of any physical field (for example, of a meson, electromagnetic, and any other such field) m a y be obtained from the variational principle which is invariant with respect to Lorentz transformation

(1)

5fL(K,. I,

...)dxdt = 0 ,

where L is the Lagrange function, and K, l, ... are field invariants. W e shall limit ourselves to equations not above the second order, and therefore K, I, ... are assumed to be composed of components of the field and its first derivatives. Then the fields arising from equation (1) h a v e the following form (*): (2)

A~-~-~2B~-

C~-~-~D

=0.

With the aim of correspondence to non-linear theory, the Lagrangian s h o u l d b e selected in such a way t h a t in the region where l~01<< ~o, I /xl, [~rf/~t I <<%/s0, the equations become linear. This is a v e r y slight limitation on the selection of the value L. L e t us now consider the velocity of signal propagation in nolo-linear theory, t h a t is the velocity of the f r o n t of a wave b o u n d e d b y a weak discontinuity. T h e value of this velocity ~ is equal to the slope of the characteristics and is determined from t h e equation: (3)

A~ ~ - 2 B ~ @

C=0.

I t is obvious t h a t this velocity will be a function of the field cp and of its ~ W i t h respect to the velocity of signal propagation, i n v a r i a n t equations m a y b e divided into two classes: A) racteristics of which do not differ from the characteristics t~1----1; B) those where l~I is in general not equal to 1. To the first class belongs for example an equation of

(4)

at:

~

all the Lawrenceequations, the chaof linear equations the following t y p e :

+ D@) = 0 ,

(*) We limit ourselves to the one dimensional case, which is entirely sufficient for our purposes.

THE

NON-LINEAIr

FIELD

THEORY

AND THE

TI~EORY

OF R E L K T I V I T Y

631

considered b y SC~IFF in a n u m b e r of his works [1, 4]. To the second class belong She equations of the electromagnetic field, equations which were suggested long ago b y M. B o ~ [2]. These equations follow from the variational principle (1) when K = 89 ~ - H ~) and I = (ell) ~. F o r the one. dimensional e~se these equations are:

(5)

(1 + ~ ) ~ + otsH ~ - -

~H-~-

+ ( 1 - - ~H ) ~ x = 0 ,

De ~H a-~+~ =o,

,(,5')

c = 1 is the velocity of light in the vacuum, and a = ( ~ L / 3 K ~ ) / ( ~ L / ~ K ) . ttEISEN]~EI%G [3] recently considered a similar equation for the meson field ~. I t should be n o t e d t h a t the signal velocity ~ of these equations is obtained in t h e following form: ,(6)

~ = ! "~/:1 + 2 o ~ K - - ~ z p q . (1 + ~p~)

I n the case of an electromagnetic field K = 89 2 - H 9 , p = s, q = H . In the case of a meson field K = 89 p = ~q~/~t, q = ~r and ~ m a y b e either less t h a n the velocity of light in the v a c u u m or it m a y be greater.

:3. - Equations of Class A. l~on-linear equations having characteristics ~ = • 1 naturally represent a t h e o r y entirely compatible with the t h e o r y of relativity. There m a y also b e found variants (the selection of the function L) which give the limited selfe n e r g y of the particles. Therefore, such equations m a y serve as a basis for a classical non-linear field theory. These equations are also compatible with the usual rules of quanVzation:

(7)

[~(x, t), ~(x', t)] = i~(x-- x').

Nevertheless, they do not lead to a consistent quantum field theory t h a t does not contain divergencies. The fact of the m a t t e r is t h a t the zero energy E o of the field, in the ease of non-linear theory, is not only infinite (as is the case also in linear theory), b u t is also non-additive to the energy of the excited s~ates. I n other words, the energy of a non-linear field cannot be represented as the sum of the energy of excitation ~ and the energy of the v a c u u m E o.

632

D.I. BLOHINCEV

F o r this reason ScmF~ introduced a m e t h o d of quantization of a non-linear field, the essence of which consists in substituting t h e continuous functions ~(~) b y the assembly of values ~0s in t h e nodes of a certain spatial lattice with a period 1 [4]. I n the linear theory, this m e t h o d would not be of a n y f u n d a m e n t a l importance, since all the results of the t h e o r y would be retained. I n the non-linear theory, the self-values of the energy of a field t e n d to c~ as exp [-- 3 ( n - - 2 ) / ( n + 2 ) ] for D ~ ate. Whereas when 1 r 0, from the equations of the field ,,

(8)

~U

w + ~ = ~,~ZAs~V~

where 9T

(9)

A~, ---- ~

it follows that, if ~ =

(lo)

ar'~(t),

e

then

G(+o)--G(-

o) -: A~a, r o ,

which signifies t h a t the signal propagates through the lattice with an infinitely large velocity (though its intensity diminishes as the distance becomes larger t h a ~ l Is - - r I). H e n c e it follows t h a t : either the self-values of the field diverge ( s - + c~) or the t h e o r y becomes incompatible with the t h e o r y of relativity (the velocity of the signal I~]> 1). T h a t is, the t h e o r y acquires features of the non-local field theory. This, however, does not discredit the non-linear equations in themselves, since we m a y still count on the application of renormalization. F o r example, the t t a m i l t o n i a n :

(11)

tt = fdx { ~ § 89

§ 89

+ ~x*q~,},

m a y be substituted b y

(11')

//* = fdx{-iz ~ + 89 ~ + ~x~rp~ + ~x*(q~*--A) ~} - - E o ,

where A and E o are renormalizing constants. Following this course, we m a y obtain a p p r o x i m a t e self-values H*, if we borrow the value ~ from linear t h e o r y ;

THE

NON-LINE_&R FIELD

THEORY

AND

THE

THEORY

OF RELATIVITY

633

we obtain:

H* ~-- ~hwknk ~- ~h2k--Eo,

<]_2)

7c

~

= e~ k~ + e~x~[1 + ~ h ~ ( n

k/ ~ (D ~)], ~ ~

= ~/c~k~ + c ~ .

k

I t follows from the above t h a t non-linear field t h e o r y does not of itself eliminate q u a n t u m divergencies, und requires additional procedures of the renormalization type.

4. - E q u a t i o n s of Class B .

The propagation of the field in this ease is similar in several respects to h y d r o d y n a m i c s on the one hand, and to crystal optics on the other. The possible equations in this class m a y be divided into two groups: t) I~1~ 1; and 2) I~t m a y also be greater t h a n i (*). The equations of the first group do not contradict the t h e o r y of relativity, b u t of course lead to the same difficulties as the equations of class A do. ]?or example, the ~ . B o ] ~ (+) equations belong to this class. I t should be noted t h a t the application of these equations to an electromagnetic field is objected to because it is proved t h a t ~he velocity of propagation of electromagnet ie signals close to the charges is not equal to the velocity of light. Therefore, the Einstein determination of time near the particles becomes invalid. One might expect t h a t it would be possible to construct such a metrics t h a t would preserve the relativistic requirement of the velocity of light being constant. However, such a metrics actually proves non-unique, since (as is the ease in crystal optics) there is not one b u t several velocities of propagation of luminous signals. I n a general ease, there are also directions of propagation for which the characteristics are imaginary [6]. The second group of equations, in which the velocity of signal propagation m a y be greater t h a n the velocity of light in the vacuum, has some features of t h e non-local t h e o r y avd is incompatible with t h a t form of causality on which the p h y s i c a l interpretation of the t h e o r y of relativity is essentially based. Thus, a very curious situation arises: the Lorentz invariancy of the variu-

(*) A curious case is possible where the equation may become elliptic [5]. (+) V. V. 0RLOV showed [6] that the Lagrangian given by M. BORN is the only one which does not lead to the formation of field shock waves.

634

n.i.

BLOHINCEu

tional principle proves insufficient b y itself for the compatibility of the field t h e o r y with the t h e o r y of relativity. I t is still necessary to require t h a t the propagation velocity of weak discontinuities should always be less than the velocity of light in the vacum. Thus we see t h a t non-linear field theories of the class B contain f u n d a m e n t a l difficulties prior to quantization, and t h a t not a n y formally invari~nt non-linear field theory is compatible with the t h e o r y of relativity. E v e n in his first work dealing with the t h e o r y of relativity, A. EI~S~EI~ left the question of simultaneousness ~ in a point ~ for f u t u r e investigators. I t is h a r d to say whether the time has come for a critical revision of t h e conceptions of space, time and causality in the region of small scales. A t a n y rate, if we take non-linear field theories seriously, t h e y bring us right up to, these problems.

REFERENCES [1] [2] [3] [4] [5] [6]

L. M. W. L. D; D.

SCHIF~: -Phys. Rev., 84, 1 (1951). B o ~ : Proc. Roy. Soc., A I43, 410 (1934). ttEISW~B~G: Zeits. /. Phys., 133, 65 (1952). SCruFF: Phys. Rev., 92, 766 (1952). BLom~exv: Dokl. Akad. Nauk SSSR, 32, 553 (1951). BLO~INC~V and u 0RLOV: Zu. j~ksper. Teor. ~iz., 25, 503 (1953).