NOETHER'S THEOREM AND GAUGE GROUPS

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1 MAR 1965 % ncrc.nc.inuq IC/65/20 INTERNATIONAL ATOMIC ENERGY AGENCY INTERNATIONAL CENTRE FOR THEORETICAL PHYSICS NOETHER'S THEOREM AND GAUGE GROUPS C. A. LOPEZ 1965 PIAZZA OBERDAN TRIESTE

2 IC/65/2O INTERNATIONAL ATOMIC ENERGY AGENCY INTERNATIONAL CENTRE FOR THEORETICAL PHYSICS t NOETHER'S THEOREM AND GAUGE GROUPS C. A. Lopez TRIESTE February 1965 To be submitted to "Nuovo Cimento" On leave of absence from Departamento de Fisica, Facultad de Ciencias, Universi&ad de Chile, Santiago, Chile

3 ABSTRACT It is shown that the usual association of a conservation law to a continuous symmetry group given "by Uoether's theorem is only correct when the group can "be defined by means of a finite number of parameters. This is the case, for instance, of the inhomogeneous proper Lorentz group and of the phase groups-. The meaning of Noether's theorem for those groups, like the gauge groups, that cannot be expressed with a finite number of parameters is discussed in detail. In addition to the well-known gauge transformations of the first and second kind, more general gauge groups of the Yang-Mills type are also considered.

4 THi^OREK AKD GAUGE GROUPS Introduction Ib is a general statement that in every case when we hav, invariance of a lagrangian under any symmetry group it follows that some physical quantity is conserved. For instance, invariance under rotations implies the conservation of angular momentum. In the particular case of continuous groups this is supposed to be a straightforward consequence of Noether's theorem.! However this is only true when the group can be described by means of a finite number of parameters. In fact Noether's theorem says that to every continuous symmetry group, i.e., a group of transformations that leave the equations of motion invariant in form and can be developed by the iteration of infinitesimal transformations, is associated a diversrence- L (1) less four-vector JV defined by J where (1.2a) are the infinitesimal transformations of the coordinates and of the field functions. The index "i" refers to the n components of the field Q.. It is a common mistake to believe that eq. (l.lb) is the differential expression of a conservation law. ^ This would be true if ]L were only a function of Q i and. ^ Qr t but actually it also depends on the symmetry group considered. Nevertheless there exists a very important class of transformations where a conservation law can be obtained from eq. (l.lb -i-

5 namely when the group depends on a finite number of parameters. The variations (1.2) may be written for this kind of group in the form 1.3a) where 6 Oi,; (j =. 1,2,, N) stands for the N parameters of the infinitesimal transformation. relations eqs. (l.l) can be written in the form On making use of these Q A j - o There are as many conservation laws as the number of parameters. The ten-parameter inhomogeneous proper Lorentz group belongs zo this type and eq. (1.4) gives the well-known external conservation laws asaociated to Lorentz invariance, i.e., energy, momentum, total angular momentum and "centre of mass' 1 theorem. Another example is the phase group defined by ' constant) U * 5) which is a one-parameter group, the parameter being/l. The corresponding conservation law is charge conservation..observe that the transformation (1.5) does not affect the coordinates ( 0/(j) 0 ), This is then an internal symmetry group. On the other hand there are other groups, notably the gauge groups, that cannot be defined by means of a finite number of parameters. The reason is that their elements depend on arbitrary functions of the coordinates./y(x). We can, for instance, define_/l (x) through a power series and take the coefficients of this series as the group parameters. Eq. (1.4) -2-

6 is no longer valid for these groups, but eq. (l.l) is still correct, In this paper we discuss the meaning of Noether's theorem when applied to several kinds of gauge groups. We find it convenient to make here a classification of these groups in three categories. The first one is the group of gauge transformations of the first kind which is a generalization of the phase group given by eq.. (l.5), namely ^ \ where the phase_/\_ (x) is an arbitrary real function. The second group is formed by the gauge transformations of the second kind of a vector field rip, U) t \ ILM f (1.1) where again -il (x) is an arbitrary real function. However, if we take into account that bothp (x) and/i,(x)iiiust satisfy field P equations that are second order partial differential equations, we conclude thatw ( x ) must have at least continuous partial derivatives up to the second order andi^fx) up to the third. Finally we call combined gauge group the set of transformations: \ (1.8) where the arbitrary real function./l (x) must have partial derivatives up to the third order. In Section 2 we consider a free vector field undergoing the transformations (l.7). Section 3 is devoted to the study of interactions between an arbitrary complex field and an arbitrary vector field with a lagrangian invariant under the 3

7 equations of motion \ = 0 combined gauge group (1.8). In Section 4 this analysis is extended to the problem of interactions of the generalized Yang-Mills type where non abelian gauge groups are considered. Finally in Section 5 w include interactions between a complex and a scalar field. 2_. The free vector field Let us consider a completely arbitrary vector field undergoing gauge transformations of the second kind (1.7). Due to the group property we can assumexl ( x ) ^s a^ infinitesimal function, because every finite transformation can be obtained as a superposition of infinitesimal transformations. We assume as usual the lagrangian density is a functional only of A^ and cl flfl, from which, on account of the action principle, we get the Now we apply Noether's theorem to the lagrangian (2..l) supposed to be invariant under the gauge group (1.7). Because this is an internal symmetry group ( QKy'D) eq.. (l.l) takes the simple form (2.3) where (2.4) from which we find that -4-

8 p ~\ eq. (2.5) Since Si. (x) is an arbitrary function we obtain from F - - F 0, (2.6a) where we have defined = (2.6c) These equations establish the existence of an antisymmetric tensor field. F n defined "by eq. (2.6c) r whose field equations are (2.6a). be interpreted as a conservation law. We emphasize that eq. (2.6a) cannot This remark is better understood when we consider the equations of motion (2.2) for 4 the vector field : (2.7) These equations show that if "the lagrangian (2.l) does not contain explicitly the field Aft r eq. (2.6a) is identical to the equation of motion (2.2). It is not a conservation law, but a true field equation. - 5 ~

9 This discussion has shown the physical meaning of 5Foether's theorem for the gauge group (1.7). The most remarkable result is that in this case it does not give any conservation law. 3. Interaction between a charged, and a vector field We consider now the total lagrangian density corresponding to two interacting fields. The first one'is a completely arbitrary complex field ^.(x), where the index "i" stands for the components (i = 1,2,....,n), and the second one is an arbitrary vector field A^ i (3.1) where j~ is the complex conjugate of j: and is considered, as usual, as an independent variable. We assume lagrangian (3.1) is invariant under the combined gauge group (l.8). We consider only infinitesimal transformations, so we can rewrite eq. (1.8) in the form (3.2) whence" the corresponding variations are A \u.* (3.3) - 6 -

10 Inserting these variations into eq. (2.3) we obtain,1 - ^ }&A'»]-o After rearrangement of eq. (3.4), we find (3-4) Defining, as usual, the current four-vector by > (3-6) we get from eq. (3-5) t n account of the arbitrariness of 3 J ^ (3.7a) (3.7b) where we have defined as in Section 2 (3.7d) Kote that these equations are not all independent. fact eq. (3.7a) is a consequence of eqs. (3.7b) and (3.7c). As a result of the application of Woether's theorem we have obtained a conserved current J source of an antisymmetric In tensor field t ^ v 7

11 We emphasize that the conservation law (3.7a) for the current arises from the fact that the phase group (1.5) is a subgroup of the combined gauge group (1.8} for the total lagrangian, namely the set of all transformations (l.8) with A ( x ) = constant. Therefore the gauge group itself does not imply a new conservation law. 4. : Generalized Yang-Mills gauge groups YA27G and KILLS introduced a three-parameter phase group to account for total isotopic spin invariance of strong interactions. They further replaced each parameter by an arbitrary real function of the coordinates (gauge group of the first kind) in order to consider invariance under local isotopic spin rotations. In analogy with electromagnetism they defined a twelve-component field (vector field with isotopic spin one) to accomplish gauge invariance. The generalization (7) to an arbitrary spin has been given by GLASHOW and GELL-MAUST. ' The underlying idea of all these attempts is to get an interaction lagrangian whenever there is some conserved (or partially conserved) quantity, in a way completely analogous to the electromagnetic theory. In this section we want to discuss some 'physical implications of these extended gauge groups with the help of Noether't theorem. This study might be of importance in the domain of strong interactions with the discovery of higher symmetries. We use the same notation as in ref. J (greek indices refer to components in coordinate space and latin indices to components in an arbitrary spin space). The gauge group is a generalization of the group (1.8) for the spinor field y (x),(r= 1,2,,N) in interaction with a 4n component field A. (oc=«1,..,4 5 i = l»2»...,n), where n is the number of gauge functions^.(x) necessary to accomplish invariance.

12 The infinitesimal transformations are w = Y r if.)-2u, M :rk AM %M, 00-7l (.(«* J X 4,k VIW 4/x) (4.1=) where M. is the rk matrix element of the spin matrix M. 1 'A / \ The constant A o is the analogous to the chargetj in (1.5; "both j\. (x) and M. are real. This is a non abelian gauge group characterised by the presence of the matrices V:, whose commutator is (4.2) where the real coefficients C. must be totally antisymmetric, They ara the familiar structure constants of the Lie algebra of the M. matrices. J Note that the non abelian nature of l the group is the reason for the appearance of the third term on the right side of eq. (4-lc). of the fields Prom eqs. (4.l) we get the infinitesimal variations (4.3a) The total lagrangian density is as usual of the form

13 If we apply Noether's theorem to this lagrangian, assumed to be invariant under the gauge group (4-l)» "we, t From eq.. (4*5) we obtain as in the previous section i - 0 (4.6) where u r Furthermore we get F - - F (4.8) (4.9) with the definition ^ Ak "*nt (4.10) x- -10-

14 Observe that the current J ^ is not only originated by the spinor field jj,. There is an additional contribution Jitrfrom the A field itself. As pointed out by YANG and this renders the equations of motion for the A field non linear, even in the absence of the spinor field. In fact we have from eqs. (4.7) to (4.10) making f' =s Q C X *)f* ~ JuY ( (4.1l) F ~ h.(.1) and the intrinsic current f is conserved on account of eq,. (4.12). / r Apart from this difference we have the same results as in Section Interaction between a charged and a scalar field We apply in this section EToether's theorem to the total lagrangian density of an arbitrary complex field in interaction with an arbitrary scalar field B. This lagrangian has the familiar form and the "gauge group" is defined by (5.2a) 6' = B + 71W (5.2c) -n-

15 whence the variations of the fields are. (5.3a) (5.3b) 8 = A M, (5.3c) As usual we impose invariance of the total lagrangian density (5.1) under the combined gauge group (5.2). We are considering this case in order to compare with the results of OGIEVETSKI and POUJBAEINOV, ^ ^ These authors claim that the minimal lagrangian compatible with the locality principle is of the form (5«l)» which they assume to be invariant under the gauge group (5-2), the introduction of a vector field being unnecessary. Application of Noether's theorem gives whence i / 'XL \u ] We emphasize an important difference between the- results of this section for a scalar field and the previous ones for vector fields, namely that in the present case the phase group (1.5) is not a subgroup of the gauge group (5*2). Therefore, invariance under this gauga group does not imply charge conservation. In fact, if we continue calling! the expression (5.6)

16 we get from eq. (5*5) -n ' (5-7) (5 " 8) On account of eq. (5-8) it turns out that eq.. (5.7) is no longer a conservation law, but an identity. Furthermore, as j cannot appear in the free lagrarsgian term for the B field we obtain from eq. (5-8) > ^ o, (5.9) (5-10) Here we cannot avoid the presence of a derivative coupling. This actually happens with the lagrangian considered by OGIEVETSKI and POLUBARIKTOV ^, which is given lay where d. y stands for the free complex field. Observe that this lagrangian is not only invariant under the gauge group (5.2), but also under the phase group (1.5), ensuring charge conservation. 6_. Conclusions We have examined several examples where the application of ffoether's theorem to gauge groups provides certain important prescriptions on the form of the total lagrangian, notably the antisymmetry of vector fields. These results we have shown to be a consequence of the structure of gauge groups, not being parametrizable with a finite number of parameters. On the other hand Lorentz group and all phase groups are parametrizable, Woether's theorem giving only the usual

17 conservation laws for them.. Gauge groups may also imply conservation laws, namely when they contain as subgroups the pararaetrizable phase groups. This is not true for the case of the scalar field considered in Section 5jOr for the free vector field, discussed in Section 2 where no. conservation law arises. One important assumption we have made in this paper is that the lagrangian must be invariant in a strict sense. actually happens with the lagrangian of electrodynamics. This Some authors do not impose invariance of the lagrangian "but only of the equations of motion. This is equivalent to saying that gauge transformations are considered as canonical transformations In that case it is enough for the lagrangian to "be invariant only up to a four-divergence. This permits the inclusion of a wider class of lagrangians, for instance those with a massive vector field. Nevertheless to accomplish gauge invariance in these cases it is necessary to add some restrictions on the gauge groups (gauge functions are no longer arbitrary, but must be solutions of a D'Alembert or Klein-Gordon equation). In other words, the physical system is only invariant under a subgroup of the general gauge group. with the strict locality principle, This is not in agreement which demands complete independence of the phase difference between two space time points. Acknowledgement The author is grateful to Professor Abdus Salam and the I.A.E.A. for the hospitality extended to him at the International Centre for Theoretical Physics, Trieste. He is also grateful to Professor Abdus Salam for some interesting comments and to Dr. Igor Saavedra for a critical reading of the manuscript. -14-

18 REFERENCES 1) P. RQMAST "Theory of Elementary Particles" (North-Holland, I960) p.220; B.L. HILL ; Rev. Mod. Phys., 23, 253 (1951). 2) E. NOETHER : Nachr. kgl. Ges. Wiss. Gottingen, 171 (1918). 4) Some authors ' call this group "gauge transformations of the first kind". This name we use only when -A. is a function of the coordinates. 5) W. N. B0G0LIUB0V and D. V. SHIRKOT : "Introduction to the Theory of Quantized Fields" (N.Y. 1959) p.19. 6) C.. YANG and R. L. MILLS : Phys. Rev. ^6_, 191 (1954). 7) " S. L. GLASHOW and M. GELL-MAM : Ann. : of Phys., ]J>, 437 (1961) 8) V. I. 0GIEVET3KI and J. V. POLUBARINOV : Muovo Cimento, 23, 173 (1962). 3) As a matter of fact eq.. (l.l) is obtained when we assume the lagrangian is itself invariant under the group considered. But the lagrangian need only be invariant up to a four-divergence. The implications of this possibility are discussed in. Section