Higher symmetry transformation of local field theories with internal symmetry

Volume 107B, number 1,2 PHYSICS LETTERS 3 December 1981

HIGHER SYMMETRY TRANSFORMATION OF

LOCAL FIELD THEORIES WITH INTERNAL SYMMETRY

Tamiaki YONEYA Institute of Physics, University of Tokyo, Komaba, Meguro-ku, Tokyo 153, Japan

Received 9 October 1981

There exist a continuously infinite number of higher symmetry transformations in general local field theories with in- ternal symmetry in arbitrary dimensions. Classically, these symmetry transformations lead to new infinite conserved currents which are nonlocal and nonpolynomial functions of the fields.

Symmetry properties are always one of the most im- portant characteristics of physical systems. In some two dimensional classical field models, infinitely many new conserved currents [1,2] have been discovered to exist. These currents are either nonlocal or nonpoly- nomial functions of the fields and suggest some higher symmetry structure of the models.

In this note, I want to point out that if one allows for nonlocality and nonpolynomiality, there in fact exist a continuously infinite number of new conserv- ed currents in general local field theories with internal symmetry in arbitrary dimensional space. These cur- rents will be derived as the Noether current correspond- ing to an infinitesimal symmetry transformation which contains several arbitrary functions as the parameters of the transformation and only changes the lagrangian density by a total divergence.

First, to illustrate the method of construction which is surprisingly simple, I take the O(N) invariant ~b 4 model in arbitrary dimensional (euclidean) space.

G ~ G ~ , + ~ 2~2 ~ x2(~) 2 O) £(X)=~ "~rrt cp a +~ .

Under a general infinitesimal variation of the field, the change of £(x) is written as

~Z(x) = G ( % G ~ % )

+ ~ba(--[-] + m 2 + X2q52)f~ba . (2)

Thus, 6£ is a total divergence if 6~ a satisfy

(-[~ + m 2 + ,'k2~b2)8~ba(X) = ~Rab(X)dPb(X), (3)

where e is an infinitesimal parameter and {Rab(X)} is an arbitrary antisymmetric real matrix function: Rab(X ) = -Rba(X ). (3) can be solved uniquely when m 2, X 2 /> 0 (otherwise, one must specify appropriate boundary conditions).

5%(x) = e f d°y(x I(-D + m 2 + X2q52) - l ly)

x G b ( Y ) % ( Y ) • (4)

Note that (4) contains ~ N ( N - 1) arbitrary functions corresponding to local O(N) rotation which is smeared out by a field dependent propagator kernel.

Since this transformation only changes the lagrang- ian density by a total divergence, it leads to the con- served Noether current.

× (x [(-r-1 + m 2 + X2¢2)- l ly)Rab(Y)q)b(y) , (5)

au/" u [x ;R] = 0 . (6)

It is now trivial to extend this construction to other models. For example, take a model o f N × N hermite matrix field M(x).

Z(x) ' = tr{g 3uMauM+ 1 2 , . 2 +Ig2M4} ~m in (7)

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Volume 107B, number 1,2 PHYSICS LETTERS 3 December 1981

An infinitesimal transformation which now contains an arbitrary hermite function V(v) and only changes (7) by a total divergence is given by

6M(x)i]

= ie fdDy(x, if 1(-[3 + m 2 + g2M2) - 1 [y, kl)

X [ V(y), M(y)] k l , (8)

where the propagator kernel is the inverse of the fol- lowing operator in the space of matrix fields.

(x, i / I ( -U + m 2 + g2M2)[y, kl) = [ ( - D + m2)SikSlt

1 2 2 + fig (g(x)ik~]l +g(x)~]6ik)] 5D(x - - y ) . (9)

The definition (9) is positive definite and also ensures the hermiticity of (8). The corresponding conserved current is

-..> .<.._

]u [x; V] = itr {M(x) (~u - ~u)

X f d D y ( x I(-V] + m 2 + g2M2) - l ly)[V(y),M(.y)] } . (10)

Similarly, the application of the method to the gauge field with the standard Yang-Mills lagrangian leads to the following equation for the infinitesimal transfor- mation of the gauge field.

(2bu(ID u, 5A~I - [D u , 6A u ] ) la

+ ig[Au,[Du,~Avl -- [Dr, 6Aul 1} = i[Vv,Av] ,

(11) where D u = Ou + igAu and no summation over v is per- formed. For (11), I have no existence or uniqueness theorem about its solution. Formally, (11) can be solved by a series expansion. A remarkable point of (11) is that it contains D arbitrary hermite functions Vv(x ) (u = 1, ..., D).

Although we have to await future elaborate study to explore about the precise nature of these higher symmetry transformations, I will mention some of their characteristics and possible physical significance of them.

(i) All of these transformations have the nature of smeared-out local internal rotation of the fields.

(ii) In the limit of large N, these transformations are always maximal [3] in the sense that the numbers of the effective degrees of freedom of the transformations

becomes asymptotically same as the numbers o f the degrees o f freedom of the field spaces. This property will be important in investigating the entropy of the field spaces in the large N limit o f matrix field theories.

I f the smearing is always short ranged in a particu- lar model, the existence of this higher transformation will lead to a finite mass gap in the model. Thus, the new symmetry transformations may in fact lead to im- portant dynamical consequences, especially, in the large N limit because of the property (ii). For exam- ple, if one could show that 5Fuv produced by the transformation (11) was short ranged on the dominant- field orbit in the large N limit [3], the area decay law of the Wilson loop could be a consequence of the higher symmetry.

In connection to these observations, it might be in- structive to discuss the Ward-Takahashi identity derived by the new transformation in the case of trivial free field theory. Take the model (1) with X = 0. Then, the Ward-Takahashi identity for the two point function (dPa(X)¢b(y)) = 6abG(X,y), takes the form

(y [([~ -- m2) -1 [z)G(x,z)

- (x I(D - m2) - l l z )G(z , y ) = 0 (12)

for arbitrary space time points x , y and z. (12) clearly implies that

G(x,y) = (x I(Vl - m2) - l ly)

apart from overall normalization constant. Therefore, the Ward-Takahashi identity corresponding to the in- variance of the action against the transformation (4) can almost replace the equation of motion. This seems to suggest that the new symmetry in general may have important dynamical contents. At present, however, it is difficult to develop the above qualitative considerations into more precise form in realistic models because of the nonpolynomiality of the transformations. An im- portant problem is thus to linearize them or to find some group theoretical structure, if any, o f the trans- formations.

I wish to thank Y. Fujii, M. Kato and M. Wadati for discussions and their interest.

[1] K. Pohlmeyer, Commun. Math. Phys. 46 (1976) 207. [2] M. Ltischer and K. Pohlmeyer, Nucl. Phys. B137 (1978)

46. [3] T. Yoneya and H. ltoyama, Univ. of Tokyo-Komaba

preprint, UT-Komaba 81-5 .

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