- Email: [email protected]
Noether's theorem in classical field theory
ANNALS
OF PHYSICS:
69, 349-363 (1972)
Noether’s
Theorem
in Classical Field Theory
JOE ROSEN Department of Physics and Astronomy, Tel-Aviv University, Tel-Aviv, Israel Received December 17, 1970 Within the lagrangian formalism in classical field theory Noether’s theorem is generalized so as to abolish the role of invariance considerations in it. Examples of application of the generalized formulation are presented for comparison with the usual formulation. It is shown that symmetry transformations do not necessarily lead to continuity equations (from which conservation laws are obtained), while transformations associated with continuity equations are not necessarily symmetry transformations. The inverse Noether theorem and families of transformations leading to the same continuity equations are discussed. Possible utility of the generalized formulation even when there is no lagrangian density is suggested.
1. INTRODUCTION The analysis presented here grew out of a dissatisfaction with Noether’s theorem as it is usually formulated [l]. This was based on the feeling that the theorem is too limited in scope and involves too much assumption. Recent work by Candotti, Palmieri, and Vitale [2] seemed to us a step in the right direction. They showed that “... Noether’s theorem can be generalized in a way that will make it clearer (if somewhat lessen) the role played by invariance considerations....” [2]. Our analysis is in this spirit and, we feel, approaches the heart of the matter. Not only do we generalize Noether’s theorem even further, but we even abolish the role played by invariance considerations. A rather amusing outcome is that symmetry transformations appear in general to be unconnected with conservation laws. Noether’s theorem usually consists of deriving a continuity equation for a current and then showing that the space integral of the time component of the current is conserved. The present analysis covers the first part of this procedure only and stops upon obtaining a continuity equation. 2. NOTATION
We use x = (x”), p = 1,2, 3,4, for space-time coordinates. Our independent fields are v(x) = (vi(x)), i = l,..., N. The summation convention holds for spacetime indices and for field component indices. We denote a = (a,) = (a/ax&) as 349 0 1972 by Academic Press, Inc.
350
ROSEN
usual and introduce d = (d,) = (d/dxLL), d2 = (d&J, etc. a, is the partial derivative with respect to xU and is concerned only with explicit x dependence, while d,, is the total derivative with respect to xU and takes implicit x dependence into consideration as well. Of course, the components of d are partial derivatives relative to each other, but in field theory the coordinate components are independent, so there is no ambiguity. We call the components of 3 partial derivatives, not to indicate their mutual partiality, but to point out their partiality relative to the other, dependent variables-the fields and their derivatives. The following examples illustrate our meaning:
&) = (au+ Mm) & + G4L4Fi)
f(x, v(x),&v(x)).
(2)
We denote the lagrangian density by 2(x, v(x), dv(x),..., &cJI(x)), where n is finite. The action functional is denoted by
Jd~l = j-, 2(x, q4%.., J’+(x)) d4x. The Euler-Lagrange E = (E”), Ed=
differential
(3)
operators for the fields v are denoted by
i (-W4,-4,(l a=0
ad
advv,, IL1*** a .
(4)
where the upper summation limit is actually the order of the highest-order field derivative among the arguments of the function upon which E acts. The first-order variation of a functionf(x, v,..., A’+) under variation of the fields and their derivatives by SF,..., &&J, respectively, is denoted by 8, f and is given by
The equality between the ftrst and second lines of Eq. (5) is not difhcult to prove. Note that the operators IP( f) are linear inf.
351
NOETHER'S THEOREM IN FIELD THEORY
3. TRANSFORMATIONS
Consider an arbitrary infinitesimal
transformation
x + x = x + 6x(x), Y-6) + F(x) = 44
+ &%
where the SF, are mutually independent,
(7)
?w, 4(x),...),
(8)
and
=wx, ~,(X>,..., d”dx)) -+ ax, 93(x),..., 44x)) = ax, yw..., 444) + QaG &),...,
J@q+)).
(9) Since the S-variations of y and 9 do not involve a change of x, it follows that a! commutes with 8. Write 89 = 4i!&Q
+ 89, )
(10) where the first term on the right hand side includes all divergence terms in 89. The action transforms as
We find that to first order
SJ = JM =
- Jv[Fl (12)
sV
[SZ] d4x,
where [S2’] = du[2’Sxu + ni“(9)
%pj + 8eQ‘] + (&pi) EiY + 86p,,
(13)
using Eq. (5). An invariance transformation is defined as a transformation under which SJ = 0 for arbitrary integration volume V. From Eq. (12) it follows that this is equivalent to [S-P] = 0.
4. EQUATIONS
OF MOTION
The equations of motion E9=0
(14)
are obtained from Eq. (13) as a necessary condition for invariance of the action for arbitrary V (i.e., [SLY] = 0) with respect to the infinitesimal transformation (7)-(lo), where: 6x = 0 for all X; 8~ arbitrary except that 89, &p),..., dn-fSrp = 0
352
ROSEN
on the boundary of V; 82 arbitrary except that 8U;U = 0 on the boundary of V and 8gz = 0 whenever v obey the equations of motion (or, more simply, just keep &Y = 0). Denote by v” any solution of the equations of motion. Since Eq. (14) is not an identity but an equation to be solved for v, it might be better written
Now for the transformation (7~(10) to be consistent, it is necessary that the transform of any solution ~JOof the original equations of motion (15) be a solution of the equations of motion in the transformed system. That is, if cpo-+ qjo = To + b Iv&J 3
(16)
EL? Iw+ = 0.
(17)
then we should have
But E~=EZ?$E85$
(18)
from Eqs. (lo), (11) and using the identity [3] Edu891u = 0.
(19)
SO by taking Eq. (18) with 9) = q” and expanding the right side about q~= q~o, we get to tist order
0 = LW + Es-%],+ (20)
with the help of Eqs. (17), (15), and (5). This is a consistency condition that must be obeyed for all solutions v” of the equations of motion (15). It is a rather messy condition. We do not make real use of it in the following.
5. SYMMETRY TRANSFORMATIONS The transformation (7)-(10) is a symmetry transformation if and only if the set of all solutions q” of the equations of motion (17) in the transformed system is identical with the set of all solutions v” of the original equations of motion (15) [4].
NOETHER'S THEOREM IN FIELD THEORY
We can express this mathematically
353
by taking Eq. (18) with v = TO:
0 = EL? lo=‘pb (21)
Using the equations of motion (15), we get the condition E 8% ImsQio= 0
(22)
for all solutions y” of the equations of motion. The consistency condition (20) then becomes
6. NOETHER TRANSFORMATIONS
In the following the notation s indicates “equality when v = ~0 for all solutions y” of the equations of motion (15).” Let the transformation (7)-(10) be such that (for a given 9’) [2]
[=I
= 42
+ k, ,
where kl@,k, are functions of x, v(x), &(x),...,
(24)
and
k, A 8Yz.
(25)
By equating Eqs. (13) and (24) we obtain d,Zu = g,
(26)
where zp = dpSX~ + nyq
&yi + 8Lzy
(27)
g = k, - (&pi) E’B - 8Li$.
(28)
up to a divergenceless vector, and
Because of the requirement equation [5]
of Eq. (25), Eq. (26) is in general a weak continuity duZu 2~ 0.
(29)
354
ROSEN
If it happens that k, = (8~~) Ei9 then Eq. (26) becomes a strong continuity d,Z” Such a transformation condition that
producing
+ &Tz,
(30)
equation = 0.
a weak continuity
(31) equation,
i.e., with the
(32) will be called a Noether transformation [6]. In the following we drop the word “weak” and use “continuity equation” to mean “weak continuity equation.” How does a Noether transformation come about? The formal expression for [&YE”],Eq. (13), contains a divergence term and a nondivergence term. When some or all of the functions appearing in these terms (8x, 8pl,%.Y, and 9) are partially or completely specified and the terms calculated, some changes might occur. The formal nondivergence term might develop a divergence part coming from (8~~) Ei.Sf’. (No divergence part can come from 8$p2, since by delmition it would be included in d,&Q.) Write this as
(33) where fi&O
(34)
due to Eq. (29, from which follows
d,fi" 2 0.
(35)
In addition, the vector of the formal divergence term might develop a divergenceless part. Express this as
where d,‘h,,“ = 0.
(37)
Then Eq. (24) takes the more detailed form [=I
= d,(hp + fi”) + fi + 8% .
(38)
NOETHER’S
THEOREM
IN
FIELD
355
THEORY
Now equate Eqs. (38) and (13) to obtain Eq. (26), where, using Eqs. (36), (37), zu = $4 8X” + P(9)
SC&+ szp - (hp +fip)
= hoi” -fiLL
(39)
up to a divergenceless vector, and g = fi - (SCJJJE”8.
(40)
So Eq. (26) reduces to dufill = &pi) Ei9
fi .
-
(41)
This equation is the essence of the whole affair and is the key to the association of continuity equations with transformations within the lagrangian formalism. Stated in words, our result is that, for a lagrangian density Y and an infinitesimal field transformation (8), if (SC& EC2 can be separated into a divergence term and a (possibly vanishing) nondivergence term as in Eq. (33) with the condition of Eq. (34), then there is associated the continuity equation (35). This condition is necessary and sufficient for a transformation (7)-(10) to be a Noether transformation. Note that Sx and &? are completely irrelevant to the matter (except that it is often 69, = f+) - 9J(x> that is given, and Scp must then be found through the relation
(42)
SC?? = scp - (6x11)d&l
(43) in practice). Also, no reference to invariance of the action nor to form invariance of the lagrangian density is made. In the usual formulation of Noether’s theorem it is assumed that the action is invariant for arbitrary integration volume, which implies [SLY] = 0, and that the lagrangian density is form invariant up to a divergence, i.e., S& = 0. Equation (38) then gives dufi”
= -d,hp
(44)
and
fi = 0,
(45) so that (8~~) EiY has no nondivergence part. Equations (33), (35), and (36) then give for the continuity equation dJ9
8x“ + nt“(c9)
8qi + 8Zlu]
=
-(&)
& 0, which is the usual result.
Ei9
(46)
356
ROSEN
When we start with 8~ and 9 more or less specified as the case may be and attempt to cast (8~~) ES2 in the form of Eqs. (33) and (34), we might find that this is always or never possible, if 8~ and 9 are sufficiently specified, or that this is conditionally possible. In the conditional case the condition, in the form of functional equations involving 8~ and/or 9, supply additional specifications needed to derive a continuity equation. Since separation into divergence and nondivergence terms is not in general unique, different continuity equations with correspondingly different conditions on sg, and/or 2 are obtainable. 7. EXAMPLES We briefly present four examples for the purpose of illustrating our approach. In each example 9 is initially unspecified, except that we assume 2’ = L?(F, dy), so that
Ei9 = (a9jayi) - du(a9jaduyi).
(47)
Each example deals with a certain transformation, so 8~ [or, equivalently, 6~ and 6x, from which 8g, is found by Eq. (43)] is given. (8,J Ei.9 is then calculated and put into the form of Eq. (33), where the divergence term is chosen so that the usual continuity equation associated with the transformation would be obtained as Eq. (35) if Eq. (34) were satisfied. Equation (34) is then a necessary and sufficient condition on 9 for the transformation to be Noether. For comparison, the usual procedure for deriving a continuity equation is to assume 6J = 0 for arbitrary V (equivalent to [8$P] = 0) and form invariance of 9 up to a divergence (i.e. &Z2 = 0) and thus to obtain Eq. (46). Then 6x, 8~, 8Z1u are plugged into Eq. (46) to obtain the corresponding continuity equation. Invariance of the action imposes a condition on 2% which is a sufficient condition for validity of the procedure. We note that s&p
= S&p, + (6X”) 4QJ = 4&p + @xv)44#
(48)
from Eq. (43) and the commutativity of 8 and d, and that 69 - &!Z’, the part of the variation of Y caused by the variations 6x and &J, is given by
S9 - l2.Y = (a2qavi) &pi + (a2qaduvi) 6dfiqli
(49)
in these examples, while from Eq. (5) 6~9
=
ww94
s9i+
(a~/a4qd4bi.
(50)
NOETHER’S
Example
THEOREM
IN
FIELD
THEORY
357
1. Local Field Transformations
where E is an infinitesimal
parameter and ri* is a constant matrix. Then
[compare with Eq. (5)], where J” = i(adpli3dp~i) rijqj
(53)
(up to a divergenceless vector) is the usual current associated with this transformation. The necessary and sufficient condition on 8 is s,6p 2%0.
(54)
The sufficient (but not necessary) condition s,z
= 0
(55)
means that 22’ is invariant under variations 8~ of Eq. (51). (In the usual formulation of Noether’s theorem condition (55) assures invariance of the action.) Example 2.
Space-Time
Translations 6x = E,
where E = (@) is an infinitesimal transform as scalars
(56)
vector parameter.
The fields are assumed to
sp, = 0,
(57)
so that by Eq. (43) 89, = -&,p
(58)
Then (&pi) Ei5? =
l ‘&9-‘~,,
(59)
where
r-", = (as~adu~ipv~i
- i+p
(60)
(up to a divergenceless vector) is the usual energy-momentum density tensor. In this example no additional condition on 9 is necessary; the transformation characterized by eq. (58) is already Noether. (Our initial assumption that 0 has no explicit x-dependence makes this possible, and in the usual Noether procedure this is what makes the action invariant.)
358 Example 3.
ROSEN
Dilations 6x = px,
where p is an infinitesimal
(61)
parameter. The fields are assumed to transform as
where DC is a constant diagonal matrix of which each diagonal element is the dimension (relative to x) of the corresponding field. Equation (43) gives (63) Then @vi) EiB
= -p&J,
+ (69
- 89 + 4p~),
(64)
where
(up to a divergenceless vector) is the usual dilation current. Yu, is the energymomentum density tensor from Eq. (60). The necessary and sufficient condition on 9 is 69 -
lL!z + 4pY z 0.
w9
The sufficient (but not necessary) condition &.P - 89 = -4pP
(67)
[compare with Eq. (62)] means that 9 is a function of dimension -4. (In the usual formulation of Noether’s theorem this assures invariance of the action.) Example 4.
Lorentz Transformations 6x” = auvx Y,
where u”yis an infinitesimal to transform as
c+ = -uvu,
antisymmetric atpi
=
Z?@V z =
(68)
tensor parameter. The fields are assumed
~u,y.z~p)j, -.p*
(6%
t 9
where the spin matrix @“’ is constant. From Eq. (43) (70)
NOETHER’S THEOREM IN FIELD THEORY
359
Then (&pi) Ei8
= -&~,a!,
J%+P + (89 - Sdp),
(71)
where (72) (up to a divergenceless vector) is the usual angular momentum density tensor. Yuu the energy-momentum density tensor from Eq. (60). The necessary and sufficient condition on 8 is SY - SY 22 0. (73) The sufficient (but not necessary) condition ss-SLY=0 means that 9’ is constructed of Lorentz invariants. (In the usual formulation is what assures invariance of the action).
(74) this
8. NOETHER TRANSFORMATIONS AND SYMMETRY TRANSFORMATIONS We have so far failed to find an esthetically pleasing (to us) connection between symmetry transformations, as characterized by Eqs. (22) and (23), and Noether transformations, as characterized by Eqs. (33) and (34), if such a connection exists at all. In fact it would seem that it does not, as indicated by the result that Eq. (22), a condition on &PZ , is all it takes [in addition to the consistency equation (20)] for a transformation to be a symmetry transformation, while %X2 is irrelevant to a transformation being Noether or not. The Venn diagram in Fig. 1 depicts the situation.
FIG. 1. RelationamongNoethertransformations (iV) andsymmetrytransformations (s) for a given lagrangiandensity.
360
ROSEN
The following is an example of a symmetry transformation a Noether transformation. Take
that is not necessarily
86p, = ES,
(75)
where E is an infinitesimal parameter. Then Eq. (22) is satisfied, and this is a symmetry transformation. For a given 9 let 8~ be such that the consistency condition (23) is satisfied. Then we can see no reason for Eqs. (33) and (34) to hold necessarily and for this transformation to be Noether.
9. NOETHER FAMILIES AND INVERSE NOETHER THEOREM A ðer fundy [6] for a given lagrangian density is the set of all transformations (7)-(10) corresponding to the same continuity equation. If 8~ is such that Eqs. (33) and (34) hold, so that Eq. (35) is the corresponding continuity equation, then an obvious way of grinding out members of this Noether family is to take various 6x’s and &“s with the same 8~. Members of this Noether family having different 8q?s can be obtained from the original 8q1 by taking [7]
where p is a function of x, v(x), d?(x),..., does not contain a divergence term, obeys p z 0,
(77)
but is otherwise arbitrary. Among these are symmetry transformations, where &KS is chosen to obey Eq. (22). We do not claim that all members of a Noether family are found by this procedure. Every Noether family contains invariance transformations as members, i.e., transformations for which [&Y] = 0. Starting with any member characterized by 6x, SF, and 89, such that Eqs. (33) and (34) hold, an invariance transformation characterized by 8x’, ST’, and 89” can be constructed as follows [8]:
89, = SqJ, &9?;’ = 8&u(arbitrary),
(78)
SLY.. = -J$ . That this is an invariance
transformation
can be seen from Eqs. (36)-(38). If
fi = 0 for &J, then OEOis form invariant up to a divergence under this invariance
NOETHER'S THEOREM IN FIELD THEORY
361
transformation, and we have the type of transformation used in the usual formulation of Noether’s theorem. The inverse Noether theorem [9] associates (for a given 94) a Noether family with a given continuity equation (35). Any transformation with [lo] sq% = (l/N)(dJ,~/PP)
(79)
corresponds to the given continuity equation, and additional members (including symmetry transformations and invariance transformations with and without invariance of L? up to a divergence) of this Noether family can be constructed as prescribed above. The situation can be depicted by the diagram of Fig. 2.
FIG. 2. Relation among Noether transformations classified according to associated continuity equation (Nx , Nz ,...), symmetry transformations (S), and invariance transformations with form-invariant (up to divergence) lagrangian density (Z) for a given lagrangian density.
lo. SOURCES
When (8~~) Ei5? separates as in Eq. (33), but Eq. (34) does not hold, we obtain from Eq. (41) a divergence equation with source [l l]
11. NOETHER WITHOUT LAGRANGIAN When, for whatever reason, use of a lagrangian density is avoided and the dynamics is described directly by equations of motion, the formulation of Noether’s theorem developed above can still be useful for obtaining continuity equations.
362
ROSEN
The dynamical evolution of the system of N fields is expressed by N independent equations of motion [12] of the form Fyx, q(x), Q+(x) ,...) 2%0. For a transformation (8) we form (&)Fi divergence terms, if possible, &r)
(81)
and separate it into divergence and non-
F’ = 42
+ fi .
(82)
If& obeys
fi sL 0, we obtain the continuity
Iffi
(83)
equation
does not obey Eq. (83), we have the divergence equation with source
ACKNOWLEDGMENTS I would like to thank the theoretical high-energy group of Chalmers Institute of Technology and especially Professor N. Svartholm and Dr. J. G. Nagel for their hospitality during the summer of 1969, when the seeds of this work were sown, and Dr. A. Rangwala and Dr. A. Joseph for discussions.
REFERENCES 1. Rather than cite more than a score of references relevant to Noether’s theorem, we refer to C. PALMIER~ AND B. VITALE, Nuovo Cimento A 66 (1970), 299, where these references are presented. We also point out the very clear article by E. L. HILL, Rev. Modern Phys. 23 (1951), 253, where a few more references are given. The role of invariance considerations in conventional formulations of Noether’s theorem is investigated in J. Rosen, Internat. J. Theoret. Phy~. 4 (1971), 287. 2. E. CANDO~~~, C. PALMIERI, AND B. VITALE, Nuovo Cimento A 70 (1970), 233. 3. This can be proved by generalizing the proof in R. COURANT AND D. IImmXRT, “Methods of Mathematical Physics” Vol. 1, p. 194, Interscience Publishers, Inc., New York, 1953. By the way, generalization of a statement on p. 195 of this book gives the following point, which, it seems, is not always appreciated: Given a divergence d,fi which is a function of x, dx),..., d”dx), the vector function f fi can be taken to be a function of x, y(x)...., d%(x), but not in general only of x, v(x) ,..., @-lp(x).
NOETHER’S
THEOREM
IN
FIELD
THEORY
363
4. A sufficient condition for this is that .Y be form invariant, L? = LZ’(8.Y = 0). A weaker sufficient condition is that the equations of motion be form invariant, E=.!? = EZ’(kZ’z = 0), so that 3’ is form invariant up to a divergence. The condition stated here is both necessary and sufficient; it can be taken as a definition of a symmetry transformation. 5. A weak continuity equation is one holding when the equations of motion are satisfied, but not in general otherwise. A strong continuity equation holds whether the equations of motion aresatisfiedorn0t.P. G.BERGMANNAND R.SCHILLER,P~~~. Rev. S~(~~~~),~,J.N.GOLDBERG, Plzys. Rev. 89 (1953), 263, AND J. C. FLETCHER, Rev. Modern Phys. 32 (1960), 65, discuss these two kinds of continuity equations with regard to deriving conservation laws from them. The weak continuity equations are what we are interested in. We mention only this: In deriving conservation laws from continuity equations (by integrating over a cylindrical 4-volume and so on) it is necessary to make assumptions about the behavior of functions of the fields and coordinates at spatial infinity. These assumptions are of a physical nature and can only be justified for solutions of the equations of motion (and even then possibly not for all solutions). 6. We take this term from Ref. [2]. 7. This is a formal construction. To ensure its existence an additional assumption about the behavior of p for ‘p --f ‘p” is necessary. 8. This is also a formal construction. 9. Many references on the inverse Noether theorem are given by Palmieri and Vitale [l]. See also Ref. [2]. 10. This is a formal construction. Concerning the existence of such expressions see H. Steudel, Friedrich-Schiller-Universitat dissertation (Jena, 1966) and Ann. Physik 20 (1967), 110. 11. The current Fiji is then of course not conserved; f2 describes the degree and manner of its nonconservation. 12. We ignore pathological cases such as dependent, too few, or too many equations of motion.
595169/=5
OF PHYSICS:
69, 349-363 (1972)
Noether’s
Theorem
in Classical Field Theory
JOE ROSEN Department of Physics and Astronomy, Tel-Aviv University, Tel-Aviv, Israel Received December 17, 1970 Within the lagrangian formalism in classical field theory Noether’s theorem is generalized so as to abolish the role of invariance considerations in it. Examples of application of the generalized formulation are presented for comparison with the usual formulation. It is shown that symmetry transformations do not necessarily lead to continuity equations (from which conservation laws are obtained), while transformations associated with continuity equations are not necessarily symmetry transformations. The inverse Noether theorem and families of transformations leading to the same continuity equations are discussed. Possible utility of the generalized formulation even when there is no lagrangian density is suggested.
1. INTRODUCTION The analysis presented here grew out of a dissatisfaction with Noether’s theorem as it is usually formulated [l]. This was based on the feeling that the theorem is too limited in scope and involves too much assumption. Recent work by Candotti, Palmieri, and Vitale [2] seemed to us a step in the right direction. They showed that “... Noether’s theorem can be generalized in a way that will make it clearer (if somewhat lessen) the role played by invariance considerations....” [2]. Our analysis is in this spirit and, we feel, approaches the heart of the matter. Not only do we generalize Noether’s theorem even further, but we even abolish the role played by invariance considerations. A rather amusing outcome is that symmetry transformations appear in general to be unconnected with conservation laws. Noether’s theorem usually consists of deriving a continuity equation for a current and then showing that the space integral of the time component of the current is conserved. The present analysis covers the first part of this procedure only and stops upon obtaining a continuity equation. 2. NOTATION
We use x = (x”), p = 1,2, 3,4, for space-time coordinates. Our independent fields are v(x) = (vi(x)), i = l,..., N. The summation convention holds for spacetime indices and for field component indices. We denote a = (a,) = (a/ax&) as 349 0 1972 by Academic Press, Inc.
350
ROSEN
usual and introduce d = (d,) = (d/dxLL), d2 = (d&J, etc. a, is the partial derivative with respect to xU and is concerned only with explicit x dependence, while d,, is the total derivative with respect to xU and takes implicit x dependence into consideration as well. Of course, the components of d are partial derivatives relative to each other, but in field theory the coordinate components are independent, so there is no ambiguity. We call the components of 3 partial derivatives, not to indicate their mutual partiality, but to point out their partiality relative to the other, dependent variables-the fields and their derivatives. The following examples illustrate our meaning:
&) = (au+ Mm) & + G4L4Fi)
f(x, v(x),&v(x)).
(2)
We denote the lagrangian density by 2(x, v(x), dv(x),..., &cJI(x)), where n is finite. The action functional is denoted by
Jd~l = j-, 2(x, q4%.., J’+(x)) d4x. The Euler-Lagrange E = (E”), Ed=
differential
(3)
operators for the fields v are denoted by
i (-W4,-4,(l a=0
ad
advv,, IL1*** a .
(4)
where the upper summation limit is actually the order of the highest-order field derivative among the arguments of the function upon which E acts. The first-order variation of a functionf(x, v,..., A’+) under variation of the fields and their derivatives by SF,..., &&J, respectively, is denoted by 8, f and is given by
The equality between the ftrst and second lines of Eq. (5) is not difhcult to prove. Note that the operators IP( f) are linear inf.
351
NOETHER'S THEOREM IN FIELD THEORY
3. TRANSFORMATIONS
Consider an arbitrary infinitesimal
transformation
x + x = x + 6x(x), Y-6) + F(x) = 44
+ &%
where the SF, are mutually independent,
(7)
?w, 4(x),...),
(8)
and
=wx, ~,(X>,..., d”dx)) -+ ax, 93(x),..., 44x)) = ax, yw..., 444) + QaG &),...,
J@q+)).
(9) Since the S-variations of y and 9 do not involve a change of x, it follows that a! commutes with 8. Write 89 = 4i!&Q
+ 89, )
(10) where the first term on the right hand side includes all divergence terms in 89. The action transforms as
We find that to first order
SJ = JM =
- Jv[Fl (12)
sV
[SZ] d4x,
where [S2’] = du[2’Sxu + ni“(9)
%pj + 8eQ‘] + (&pi) EiY + 86p,,
(13)
using Eq. (5). An invariance transformation is defined as a transformation under which SJ = 0 for arbitrary integration volume V. From Eq. (12) it follows that this is equivalent to [S-P] = 0.
4. EQUATIONS
OF MOTION
The equations of motion E9=0
(14)
are obtained from Eq. (13) as a necessary condition for invariance of the action for arbitrary V (i.e., [SLY] = 0) with respect to the infinitesimal transformation (7)-(lo), where: 6x = 0 for all X; 8~ arbitrary except that 89, &p),..., dn-fSrp = 0
352
ROSEN
on the boundary of V; 82 arbitrary except that 8U;U = 0 on the boundary of V and 8gz = 0 whenever v obey the equations of motion (or, more simply, just keep &Y = 0). Denote by v” any solution of the equations of motion. Since Eq. (14) is not an identity but an equation to be solved for v, it might be better written
Now for the transformation (7~(10) to be consistent, it is necessary that the transform of any solution ~JOof the original equations of motion (15) be a solution of the equations of motion in the transformed system. That is, if cpo-+ qjo = To + b Iv&J 3
(16)
EL? Iw+ = 0.
(17)
then we should have
But E~=EZ?$E85$
(18)
from Eqs. (lo), (11) and using the identity [3] Edu891u = 0.
(19)
SO by taking Eq. (18) with 9) = q” and expanding the right side about q~= q~o, we get to tist order
0 = LW + Es-%],+ (20)
with the help of Eqs. (17), (15), and (5). This is a consistency condition that must be obeyed for all solutions v” of the equations of motion (15). It is a rather messy condition. We do not make real use of it in the following.
5. SYMMETRY TRANSFORMATIONS The transformation (7)-(10) is a symmetry transformation if and only if the set of all solutions q” of the equations of motion (17) in the transformed system is identical with the set of all solutions v” of the original equations of motion (15) [4].
NOETHER'S THEOREM IN FIELD THEORY
We can express this mathematically
353
by taking Eq. (18) with v = TO:
0 = EL? lo=‘pb (21)
Using the equations of motion (15), we get the condition E 8% ImsQio= 0
(22)
for all solutions y” of the equations of motion. The consistency condition (20) then becomes
6. NOETHER TRANSFORMATIONS
In the following the notation s indicates “equality when v = ~0 for all solutions y” of the equations of motion (15).” Let the transformation (7)-(10) be such that (for a given 9’) [2]
[=I
= 42
+ k, ,
where kl@,k, are functions of x, v(x), &(x),...,
(24)
and
k, A 8Yz.
(25)
By equating Eqs. (13) and (24) we obtain d,Zu = g,
(26)
where zp = dpSX~ + nyq
&yi + 8Lzy
(27)
g = k, - (&pi) E’B - 8Li$.
(28)
up to a divergenceless vector, and
Because of the requirement equation [5]
of Eq. (25), Eq. (26) is in general a weak continuity duZu 2~ 0.
(29)
354
ROSEN
If it happens that k, = (8~~) Ei9 then Eq. (26) becomes a strong continuity d,Z” Such a transformation condition that
producing
+ &Tz,
(30)
equation = 0.
a weak continuity
(31) equation,
i.e., with the
(32) will be called a Noether transformation [6]. In the following we drop the word “weak” and use “continuity equation” to mean “weak continuity equation.” How does a Noether transformation come about? The formal expression for [&YE”],Eq. (13), contains a divergence term and a nondivergence term. When some or all of the functions appearing in these terms (8x, 8pl,%.Y, and 9) are partially or completely specified and the terms calculated, some changes might occur. The formal nondivergence term might develop a divergence part coming from (8~~) Ei.Sf’. (No divergence part can come from 8$p2, since by delmition it would be included in d,&Q.) Write this as
(33) where fi&O
(34)
due to Eq. (29, from which follows
d,fi" 2 0.
(35)
In addition, the vector of the formal divergence term might develop a divergenceless part. Express this as
where d,‘h,,“ = 0.
(37)
Then Eq. (24) takes the more detailed form [=I
= d,(hp + fi”) + fi + 8% .
(38)
NOETHER’S
THEOREM
IN
FIELD
355
THEORY
Now equate Eqs. (38) and (13) to obtain Eq. (26), where, using Eqs. (36), (37), zu = $4 8X” + P(9)
SC&+ szp - (hp +fip)
= hoi” -fiLL
(39)
up to a divergenceless vector, and g = fi - (SCJJJE”8.
(40)
So Eq. (26) reduces to dufill = &pi) Ei9
fi .
-
(41)
This equation is the essence of the whole affair and is the key to the association of continuity equations with transformations within the lagrangian formalism. Stated in words, our result is that, for a lagrangian density Y and an infinitesimal field transformation (8), if (SC& EC2 can be separated into a divergence term and a (possibly vanishing) nondivergence term as in Eq. (33) with the condition of Eq. (34), then there is associated the continuity equation (35). This condition is necessary and sufficient for a transformation (7)-(10) to be a Noether transformation. Note that Sx and &? are completely irrelevant to the matter (except that it is often 69, = f+) - 9J(x> that is given, and Scp must then be found through the relation
(42)
SC?? = scp - (6x11)d&l
(43) in practice). Also, no reference to invariance of the action nor to form invariance of the lagrangian density is made. In the usual formulation of Noether’s theorem it is assumed that the action is invariant for arbitrary integration volume, which implies [SLY] = 0, and that the lagrangian density is form invariant up to a divergence, i.e., S& = 0. Equation (38) then gives dufi”
= -d,hp
(44)
and
fi = 0,
(45) so that (8~~) EiY has no nondivergence part. Equations (33), (35), and (36) then give for the continuity equation dJ9
8x“ + nt“(c9)
8qi + 8Zlu]
=
-(&)
& 0, which is the usual result.
Ei9
(46)
356
ROSEN
When we start with 8~ and 9 more or less specified as the case may be and attempt to cast (8~~) ES2 in the form of Eqs. (33) and (34), we might find that this is always or never possible, if 8~ and 9 are sufficiently specified, or that this is conditionally possible. In the conditional case the condition, in the form of functional equations involving 8~ and/or 9, supply additional specifications needed to derive a continuity equation. Since separation into divergence and nondivergence terms is not in general unique, different continuity equations with correspondingly different conditions on sg, and/or 2 are obtainable. 7. EXAMPLES We briefly present four examples for the purpose of illustrating our approach. In each example 9 is initially unspecified, except that we assume 2’ = L?(F, dy), so that
Ei9 = (a9jayi) - du(a9jaduyi).
(47)
Each example deals with a certain transformation, so 8~ [or, equivalently, 6~ and 6x, from which 8g, is found by Eq. (43)] is given. (8,J Ei.9 is then calculated and put into the form of Eq. (33), where the divergence term is chosen so that the usual continuity equation associated with the transformation would be obtained as Eq. (35) if Eq. (34) were satisfied. Equation (34) is then a necessary and sufficient condition on 9 for the transformation to be Noether. For comparison, the usual procedure for deriving a continuity equation is to assume 6J = 0 for arbitrary V (equivalent to [8$P] = 0) and form invariance of 9 up to a divergence (i.e. &Z2 = 0) and thus to obtain Eq. (46). Then 6x, 8~, 8Z1u are plugged into Eq. (46) to obtain the corresponding continuity equation. Invariance of the action imposes a condition on 2% which is a sufficient condition for validity of the procedure. We note that s&p
= S&p, + (6X”) 4QJ = 4&p + @xv)44#
(48)
from Eq. (43) and the commutativity of 8 and d, and that 69 - &!Z’, the part of the variation of Y caused by the variations 6x and &J, is given by
S9 - l2.Y = (a2qavi) &pi + (a2qaduvi) 6dfiqli
(49)
in these examples, while from Eq. (5) 6~9
=
ww94
s9i+
(a~/a4qd4bi.
(50)
NOETHER’S
Example
THEOREM
IN
FIELD
THEORY
357
1. Local Field Transformations
where E is an infinitesimal
parameter and ri* is a constant matrix. Then
[compare with Eq. (5)], where J” = i(adpli3dp~i) rijqj
(53)
(up to a divergenceless vector) is the usual current associated with this transformation. The necessary and sufficient condition on 8 is s,6p 2%0.
(54)
The sufficient (but not necessary) condition s,z
= 0
(55)
means that 22’ is invariant under variations 8~ of Eq. (51). (In the usual formulation of Noether’s theorem condition (55) assures invariance of the action.) Example 2.
Space-Time
Translations 6x = E,
where E = (@) is an infinitesimal transform as scalars
(56)
vector parameter.
The fields are assumed to
sp, = 0,
(57)
so that by Eq. (43) 89, = -&,p
(58)
Then (&pi) Ei5? =
l ‘&9-‘~,,
(59)
where
r-", = (as~adu~ipv~i
- i+p
(60)
(up to a divergenceless vector) is the usual energy-momentum density tensor. In this example no additional condition on 9 is necessary; the transformation characterized by eq. (58) is already Noether. (Our initial assumption that 0 has no explicit x-dependence makes this possible, and in the usual Noether procedure this is what makes the action invariant.)
358 Example 3.
ROSEN
Dilations 6x = px,
where p is an infinitesimal
(61)
parameter. The fields are assumed to transform as
where DC is a constant diagonal matrix of which each diagonal element is the dimension (relative to x) of the corresponding field. Equation (43) gives (63) Then @vi) EiB
= -p&J,
+ (69
- 89 + 4p~),
(64)
where
(up to a divergenceless vector) is the usual dilation current. Yu, is the energymomentum density tensor from Eq. (60). The necessary and sufficient condition on 9 is 69 -
lL!z + 4pY z 0.
w9
The sufficient (but not necessary) condition &.P - 89 = -4pP
(67)
[compare with Eq. (62)] means that 9 is a function of dimension -4. (In the usual formulation of Noether’s theorem this assures invariance of the action.) Example 4.
Lorentz Transformations 6x” = auvx Y,
where u”yis an infinitesimal to transform as
c+ = -uvu,
antisymmetric atpi
=
Z?@V z =
(68)
tensor parameter. The fields are assumed
~u,y.z~p)j, -.p*
(6%
t 9
where the spin matrix @“’ is constant. From Eq. (43) (70)
NOETHER’S THEOREM IN FIELD THEORY
359
Then (&pi) Ei8
= -&~,a!,
J%+P + (89 - Sdp),
(71)
where (72) (up to a divergenceless vector) is the usual angular momentum density tensor. Yuu the energy-momentum density tensor from Eq. (60). The necessary and sufficient condition on 8 is SY - SY 22 0. (73) The sufficient (but not necessary) condition ss-SLY=0 means that 9’ is constructed of Lorentz invariants. (In the usual formulation is what assures invariance of the action).
(74) this
8. NOETHER TRANSFORMATIONS AND SYMMETRY TRANSFORMATIONS We have so far failed to find an esthetically pleasing (to us) connection between symmetry transformations, as characterized by Eqs. (22) and (23), and Noether transformations, as characterized by Eqs. (33) and (34), if such a connection exists at all. In fact it would seem that it does not, as indicated by the result that Eq. (22), a condition on &PZ , is all it takes [in addition to the consistency equation (20)] for a transformation to be a symmetry transformation, while %X2 is irrelevant to a transformation being Noether or not. The Venn diagram in Fig. 1 depicts the situation.
FIG. 1. RelationamongNoethertransformations (iV) andsymmetrytransformations (s) for a given lagrangiandensity.
360
ROSEN
The following is an example of a symmetry transformation a Noether transformation. Take
that is not necessarily
86p, = ES,
(75)
where E is an infinitesimal parameter. Then Eq. (22) is satisfied, and this is a symmetry transformation. For a given 9 let 8~ be such that the consistency condition (23) is satisfied. Then we can see no reason for Eqs. (33) and (34) to hold necessarily and for this transformation to be Noether.
9. NOETHER FAMILIES AND INVERSE NOETHER THEOREM A ðer fundy [6] for a given lagrangian density is the set of all transformations (7)-(10) corresponding to the same continuity equation. If 8~ is such that Eqs. (33) and (34) hold, so that Eq. (35) is the corresponding continuity equation, then an obvious way of grinding out members of this Noether family is to take various 6x’s and &“s with the same 8~. Members of this Noether family having different 8q?s can be obtained from the original 8q1 by taking [7]
where p is a function of x, v(x), d?(x),..., does not contain a divergence term, obeys p z 0,
(77)
but is otherwise arbitrary. Among these are symmetry transformations, where &KS is chosen to obey Eq. (22). We do not claim that all members of a Noether family are found by this procedure. Every Noether family contains invariance transformations as members, i.e., transformations for which [&Y] = 0. Starting with any member characterized by 6x, SF, and 89, such that Eqs. (33) and (34) hold, an invariance transformation characterized by 8x’, ST’, and 89” can be constructed as follows [8]:
89, = SqJ, &9?;’ = 8&u(arbitrary),
(78)
SLY.. = -J$ . That this is an invariance
transformation
can be seen from Eqs. (36)-(38). If
fi = 0 for &J, then OEOis form invariant up to a divergence under this invariance
NOETHER'S THEOREM IN FIELD THEORY
361
transformation, and we have the type of transformation used in the usual formulation of Noether’s theorem. The inverse Noether theorem [9] associates (for a given 94) a Noether family with a given continuity equation (35). Any transformation with [lo] sq% = (l/N)(dJ,~/PP)
(79)
corresponds to the given continuity equation, and additional members (including symmetry transformations and invariance transformations with and without invariance of L? up to a divergence) of this Noether family can be constructed as prescribed above. The situation can be depicted by the diagram of Fig. 2.
FIG. 2. Relation among Noether transformations classified according to associated continuity equation (Nx , Nz ,...), symmetry transformations (S), and invariance transformations with form-invariant (up to divergence) lagrangian density (Z) for a given lagrangian density.
lo. SOURCES
When (8~~) Ei5? separates as in Eq. (33), but Eq. (34) does not hold, we obtain from Eq. (41) a divergence equation with source [l l]
11. NOETHER WITHOUT LAGRANGIAN When, for whatever reason, use of a lagrangian density is avoided and the dynamics is described directly by equations of motion, the formulation of Noether’s theorem developed above can still be useful for obtaining continuity equations.
362
ROSEN
The dynamical evolution of the system of N fields is expressed by N independent equations of motion [12] of the form Fyx, q(x), Q+(x) ,...) 2%0. For a transformation (8) we form (&)Fi divergence terms, if possible, &r)
(81)
and separate it into divergence and non-
F’ = 42
+ fi .
(82)
If& obeys
fi sL 0, we obtain the continuity
Iffi
(83)
equation
does not obey Eq. (83), we have the divergence equation with source
ACKNOWLEDGMENTS I would like to thank the theoretical high-energy group of Chalmers Institute of Technology and especially Professor N. Svartholm and Dr. J. G. Nagel for their hospitality during the summer of 1969, when the seeds of this work were sown, and Dr. A. Rangwala and Dr. A. Joseph for discussions.
REFERENCES 1. Rather than cite more than a score of references relevant to Noether’s theorem, we refer to C. PALMIER~ AND B. VITALE, Nuovo Cimento A 66 (1970), 299, where these references are presented. We also point out the very clear article by E. L. HILL, Rev. Modern Phys. 23 (1951), 253, where a few more references are given. The role of invariance considerations in conventional formulations of Noether’s theorem is investigated in J. Rosen, Internat. J. Theoret. Phy~. 4 (1971), 287. 2. E. CANDO~~~, C. PALMIERI, AND B. VITALE, Nuovo Cimento A 70 (1970), 233. 3. This can be proved by generalizing the proof in R. COURANT AND D. IImmXRT, “Methods of Mathematical Physics” Vol. 1, p. 194, Interscience Publishers, Inc., New York, 1953. By the way, generalization of a statement on p. 195 of this book gives the following point, which, it seems, is not always appreciated: Given a divergence d,fi which is a function of x, dx),..., d”dx), the vector function f fi can be taken to be a function of x, y(x)...., d%(x), but not in general only of x, v(x) ,..., @-lp(x).
NOETHER’S
THEOREM
IN
FIELD
THEORY
363
4. A sufficient condition for this is that .Y be form invariant, L? = LZ’(8.Y = 0). A weaker sufficient condition is that the equations of motion be form invariant, E=.!? = EZ’(kZ’z = 0), so that 3’ is form invariant up to a divergence. The condition stated here is both necessary and sufficient; it can be taken as a definition of a symmetry transformation. 5. A weak continuity equation is one holding when the equations of motion are satisfied, but not in general otherwise. A strong continuity equation holds whether the equations of motion aresatisfiedorn0t.P. G.BERGMANNAND R.SCHILLER,P~~~. Rev. S~(~~~~),~,J.N.GOLDBERG, Plzys. Rev. 89 (1953), 263, AND J. C. FLETCHER, Rev. Modern Phys. 32 (1960), 65, discuss these two kinds of continuity equations with regard to deriving conservation laws from them. The weak continuity equations are what we are interested in. We mention only this: In deriving conservation laws from continuity equations (by integrating over a cylindrical 4-volume and so on) it is necessary to make assumptions about the behavior of functions of the fields and coordinates at spatial infinity. These assumptions are of a physical nature and can only be justified for solutions of the equations of motion (and even then possibly not for all solutions). 6. We take this term from Ref. [2]. 7. This is a formal construction. To ensure its existence an additional assumption about the behavior of p for ‘p --f ‘p” is necessary. 8. This is also a formal construction. 9. Many references on the inverse Noether theorem are given by Palmieri and Vitale [l]. See also Ref. [2]. 10. This is a formal construction. Concerning the existence of such expressions see H. Steudel, Friedrich-Schiller-Universitat dissertation (Jena, 1966) and Ann. Physik 20 (1967), 110. 11. The current Fiji is then of course not conserved; f2 describes the degree and manner of its nonconservation. 12. We ignore pathological cases such as dependent, too few, or too many equations of motion.
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