Is the principle of least action a must?

The least action principle occupies a central part in contemporary physics. Yet, as far as classical field theory is concerned, it may not be as essential as generally thought. We show with three detailed examples of classical interacting field theories that it is possible, in cases of physical interest, to derive the correct field equations for all fields from the action (which we regard as defining the theory), some of its symmetries, and the conservation law of energy-momentum (this last regarded as ultimately coming from experiment)


I. INTRODUCTION
The principle of least action or Hamilton's principle (henceforth LAP) occupies a central position in contemporary physics. Beginning with Lagrange and Euler's particle dynamics, continuing through field theory and culminating with string theory, the equations of motion of a fundamental theory are generally derived by starting from the theory's action and then invoking the LAP. Another central pillar of physics is the Nöther principle whereby a continuous symmetry of a field's action, when combined with the field equations (themselves typically derived from LAP), yields the conservation law (continuity equation) for some continuous field quantity [1].
In simple cases the above logic can be inverted. A well known example is furnished by the theory of the pure (sourceless) electromagnetic field. One may infer its action from the requirements of gauge symmetry, time reversal symmetry and Lorentz invariance. From that action one infers an energy momentum tensor (by the well known recipes for the canonical or symmetric one). Then by requiring that the tensor's divergence vanish (energy-momentum conservation) one obtains a set of equations which, for generic values of the Faraday tensor, imply that the latter's divergence must vanish [2]. Thus the Gauss and Ampere laws emerge from energy-momentum conservation, itself a consequence of the symmetry of the action under spacetime translations. The magnetic Gauss and Faraday laws are automatic consequences of the use of the electromagnetic 4-potential as basic field variable. Thus, in the mentioned example, the symmetries plus conservation laws lead to the field equations without any appeal being made to the LAP.
Other examples are almost as easy to implement when they deal with a single field (including a self-interacting one), or with a set of noninteracting fields. In the later case separate conservation laws for energy-momentum exist for each field, and the procedure very much follows that for a single field. However, once different fields are coupled, a single conservation law of each type exists for the set of fields, and obviously supplies considerably less information than in the noninteracting case. Thus we can ask, can the field equations for each of the fields in this case always be inferred without appealing to the LAP?
The question is interesting inasmuch as it is easily transmuted into a more fundamental one. From a practical point of view one can hardly analyze a field theory, or predict with it, without knowing its field equations. But which are the primary entities essential in reaching the field equation? Is it the Lagrangian and LAP, or is it symmetries, the action and conservation laws?
In this paper we present three nontrivial examples of interacting field systems; for each we recover the correct field equations by starting from symmetries and conservation laws, and never invoking the LAP. We thus suspect that in large tracts of the field theory zoo, all field equations can be recovered from symmetries and conservation laws when judiciously applied to the action suggested by symmetries, but without need of the LAP. Symmetries are, of course, an intuitive primitive concept in physics. And it is well known how to go about formulating an action for a set of fields with specified characteristics from the requirement that the action incorporate the symmetries. In addition we can think (and in our approach we should think ) of conservation laws as experimental facts. After all the experimental verification of a conservation law is in a real sense a null experiment, and thus very accurate by nature. Thus we contemplate a way of doing physics which is conceptually simpler and with sounder experimental basis than is the traditional approach to field dynamics.
By contrast, the LAP idea is certainly distant from experiment and difficult to internalize conceptually. It is a teleological principle quite at variance with the traditional way we operate in classical physics, where initial conditions are given weight while final conditions have no part. Historically the LAP was controversial for many years, beginning with de Maupertuis's half mystic and somewhat vague formulation of the principle [3]. Thus the approach that sidesteps the LAP would seem to add clarity to physics.
But lest a misunderstanding occur we hasten to add that, as a matter of practice, the traditional way to the field equations via LAP will most probably retain its popularity. The approach via symmetries and conservation laws described here usually involves intricate arguments which are different for different fields, and would not typically be more economical than its traditional counterpart.
Of course all that has been said concerns classical physics. In quantum field theory the dynamics comes from Feynman's functional (or path) integral. The exponential in its integrand contains the field's action, so that the starting point of a quantum field theory is similar to that of its classical counterpart. Since the functional integral cannot usually be calculated exactly, one opts for an approximation scheme such as the popular loop approximation scheme. In lowest (semiclassical or tree) approximation, this scheme entails evaluating the said exponential at the extremal value of the action with initial and final conditions for the fields being specified. But this extremal value is, of course, determined by the LAP, and obtains precisely for a field configuration that evolves according to the classical field equations. Thus LAP has a natural place in the so called semiclassical approximation for the functional integral, and would seem an essential part of the quantum description.
By contrast, as we have just argued, when one is concerned solely with classical field theory, the LAP seems not to be essential.
The plan of the paper is as follows. In Sec. II we study the most general system of scalar fields in curved spacetime with minimal coupling to the metric. We show that in ddimensional spacetime the scalar field equations for d or fewer fields scan be derived without help of the LAP. In Sec. III we consider a charged scalar field with arbitrary self-potential interacting with the electromagnetic field. We demonstrate in flat spacetime that the scalar and electromagnetic field equations can be obtained from symmetry and conservation considerations without use of the LAP. In Sec. IV we take up the theory of Dirac fields interacting with a non-Abelian gauge field, typically with symmetry of SU(n). We show that the gauge field equation emerges without use of the LAP; likewise, for a representation whose dimension is no higher than half the number of generators [n 2 − 1 for SU(n)], the Dirac field equation may be obtained from symmetry and conservation considerations alone.

A. Lagrangian and definitions
In d dimensional curved spacetime let us take the action as a local, very general one, for a collection of interacting scalar fields φ µ with µ = 1, 2, · · · , N ≤ d: A trivial example would be one for which L is a linear combination of invariants of the form ∂ a φ µ ∂ a φ µ and φ 2 µ for several (noninteracting) scalar fields labelled by index µ. A somewhat more complex example would be one for two massive interacting scalar fields But, of course, the action (1) includes cases in which the Lagrangian density is not separable into kinetic and potential parts.
In terms of the definition of the functional derivative of the action we introduce the notation as well as Definition (4) does not necessarily imply that we are using the LAP. As well known [4], it emerges just as a consequence of the diffeomorphism invariance of the non-gravitational part of the action, in which one regards the metric as given, and does not enter into the question of which equations determine it. We shall employ the definition (4) throughout.

B. Consequences of diffeomorphism invariance
We now consider an infinitesimal increment of the action, δS, engendered by infinitesimal where we sum over µ = 1, · · · , N, and have used Gauss' theorem to convert a divergence into an integral over the d − 1 dimensional boundary ∂V with volume element dΣ a .
At this stage we part ways with the usual derivation via the LAP. We shall only assume that S is invariant under diffeomorphisms x a → x a + χ a (x) with the χ a being d arbitrary differentiable functions within V, but vanishing on ∂V and outside V. Under an infinitesimal diffeomorphism of this kind, implemented as a Lie drag [5,6], δ χ φ = χ a ∂ a φ as well as δ χ g ab = −(χ a;b + χ b;a ). Thus the boundary term vanishes and we can write where we have exploited the symmetry of T ab and integrated by parts by virtue of the identity and exploited the fact that χ a vanishes on ∂V. Once we take into account the conservation of energy-momentum, T ab ;b = 0, we obtain The arbitrariness of the χ a allows us to conclude that within V for every index a For a single scalar field φ this immediately implies that E φ = 0 since in the generic situation ∂ a φ 1 = 0 throughout V apart, perhaps, from special points or surfaces. By continuity E φ = 0 also at these special locations. Thus we see from Eq. (3) that the field obeys the usual field equation, and this without the LAP having been invoked. How to extend this argument to many fields?

C. The argument for several fields
Let us select N different coordinate components of Eq. (9) and label them by a = k 1 , k 2 , · · · ; each component consists of N terms. The entire set can be written in terms In this and the next three paragraph we regard those coordinates x a that are not included in {x k 1 , x k 2 , · · · , x k N } as fixed; thus we at first consider only a subspace V of the whole spacetime spanned by the {x k 1 , x k 2 , · · · , x k N }. This only coincides with the whole spacetime The determinant of the square matrix here is the Jacobian Generically J cannot vanish. For were it to vanish this would indicate that, viewed as functions of x k 1 , x k 2 , · · · , x k N , the φ 1 , φ 2 , · · · , φ N are functionally related. This, of course, will not be true in the generic physical field configuration. It can thus be assumed that J = 0, except perhaps at isolated point sets. Then the nonvanishing character of J tells us that the E φ column vector must vanish (including at the special points by the argument of continuity). But the vanishing of this "vector" is just the collection of all the Euler-Lagrange equations, Eqs. (3), for the set of fields φ µ . These field equations are applicable within the said subspace of spacetime in V.
As mentioned earlier, if N = d, V covers the whole spacetime. If N = d − 1 we can set up d distinct equations of the form (10), one for each coordinate left out the list Carrying out the above procedure using the d distinct Jacobians allows us to extend the previous conclusion to the the whole of spacetime when V itself is allowed to expand without bound. The above can be suitably generalized for N ≤ d − 2. We may thus obtain all the Euler-Lagrange field equations all over spacetime, and this without appeal to the LAP.
Of course, if the number of fields φ µ exceeds d, the above analysis is insufficient to obtain all field equations. One would then have to appeal to other symmetries.

A. Gauge invariant action
Here the theory representing a charged (complex) scalar field interacting with a SU (1) gauge field will be considered [7]. As we saw in Sec. II B the principal effect of curved space is to make a term containing the energy-momentum tensor appear under an integral when we carry out a diffeomorphism. To save labour we presume that term has been dropped by invoking energy conservation, as we did earlier. Thus we may revert to Minkowski spacetime.
We also restrict attention to four spacetime dimensions. The extension to higher dimensional flat spacetime is straightforward.
The action on the four dimensional Minkowski background is given by where the gauge covariant derivative for scalars is defined by D a ≡ ∂ a + ıeA a , and the field strength by Here A a is the gauge vector potential and e is the coupling constant. This action is gauge invariant under local gauge transformations A a → A a + e −1 ∂ a Λ, φ → φ exp(−ıΛ) and φ * → φ * exp(ıΛ), where Λ, the gauge function, is an arbitrary function of spacetime.
Next, in analogy with Eq. (3), denote the functional derivatives of the action with respect to the fields by the following symbols:

B. Scale symmetry and Bianchi identity
Now we compute the change engendered in S by any specified infinitesimal increments in A a , φ and φ * . After integrating by parts and discarding the boundary terms (the last requires that δφ, δφ * and δA a to vanish on the boundary), as done in Sec. II, we get In case one also increments parameters in the action, extra terms will appear here.
It must be noted that apart from gauge symmetry the action Eq. (12) is invariant under a particular kind of scale transformations as well as under diffeomorphisms. This information will now be put to use to find the field equations for the gauge and scalar fields.
It is easy to check that S is invariant under the scale transformation x a → εx a , A a → A a /ε, φ → φ/ε, φ * → φ * /ε and m → m/ε, with ε constant throughout spacetime. The metric is here regarded as scale invariant (the relevant scaling changes being taken up by the coordinates). We now interpret the collection of small increments mentioned earlier as due to a small scale transformation; accordinglysε = 1 + δε with δε infinitesimal. We thus In view of all these we obtain to O(ε) The boundary terms in this particular case are annulled provided that A a , φ and φ * themselves vanish on the boundary. This is certainly implementable if the boundary is taken to be at infinity.
Of course, since S is scale invariant in any gauge, Eq. (15) should hold in any gauge provided ε is the same in all gauges. Now by evaluating the explicit expressions from Eq. (13) it may be verified that E a A , E φ φ and E φ * φ * , as well as φφ * are all gauge invariant.
Likewise L from Eq. (12) is gauge invariant. But, as we know, the vector potential is not: Now requiring that δ Λ (δ ε S) = 0, discarding the boundary term (which entails requiring Λ = 0 on the boundary) and taking into account the arbitrariness of Λ everywhere else, we With the benefit of hindsight this can be recognized as a combination of a trivial identity and the conservation of charge, namely

C. Gauge symmetry and charge conservation
Let us now go back to Eq. (14) and interpret the increments mentioned there as due to an infinitesimal gauge transformation: Discarding the boundary terms entails here having both Λ and ∂ a Λ vanish on the boundary; we get Since Λ is arbitrary inside the boundary we find that everywhere in the bulk In view of Eq. (17), and the obvious fact that in a generic configuration φ cannot vanish identically, we see that To elucidate the physical content of this result let us substitute in it the explicit expressions of E φ and E φ * . From Eq. (12) we obtain After substitution in Eq. (20) and some manipulations we obtain which is recognized as the continuity equation for the conservation of U(1) charge together with the traditional expression for the current. It is remarkable that in contrast to the textbook approach, these results are here obtained without any reference to the explicit equations of motion. The analysis here is "off-shell".
We shall now investigate consequences of the coordinate invariance of the action (12).
Actually a full investigation in this direction would have to be performed in curved spacetime since the metric changes in a nontrivial way under diffeomorphisms. As in Sec. II B such change engenders a term proportional to the divergence of the energy momentum tensor.
Since our approach is to impose energy-momentum conservation, this term will drop out.
Hence we have cut corners here, and have done all the work in flat spacetime.

D. Diffeomorphism symmetry leads to field equations
We shall now interpret the increments referred to in Eq. (14) as due to an infinitesimal diffeomorphism, namely x a → x a + χ a with χ a small. As in Sec. II B we implement this by Lie dragging [5,6], to wit Substituting these together with Eq. (20) into Eq. (14) gives By calling on Eq. (17) we may complete the derivative implied by the first term here to get If we agree that the χ a all vanish at the boundary, the perfect divergence here in seen to integrate to a vanishing boundary term. In view of the arbitrariness of χ b within V we get In view of the antisymmetry of F ba contraction of this equation with E b A yields If we ignore the non-generic configuration with |φ| = const., the above equation has two solutions: either (I) E φ = 0 or (II) E b A ∂ b |φ| 2 = 0. We consider each case separately.
Case I: E φ = 0, at least over a finite spacetime region.
Of course, since φ (and consequently φ * ) cannot vanish identically, Eq. (20) immediately implies that E φ * = 0 throughout the same region. It now follows from Eq. (25) that everywhere in that region Now if we write out the determinant of the matrix made up of components F ba we see that it cannot vanish identically (its vanishing would imply a very non-generic configuration). The only solution of the linear system of d equations is thus E a A = 0. Thus, without invoking the LAP, we have have found all the equations of motion in the said spacetime region: For the generic configuration we cannot have |φ| 2 a spacetime constant. Hence E a A is orthogonal to ∂ b |φ| 2 , except perhaps on special surfaces or points. Consider now the scalar where f is an arbitrary differentiable real function of its argument. In view of Eq. (17) as well as the present assumption that E b A ∂ b |φ| 2 = 0 we see that U α = 0. Thus f (|φ| 2 )E a A is a spacetime constant in the stipulated region (this being also true by continuity over the mentioned special surfaces). Obviously, because of the arbitrariness of the form of f (x), this can be true only if E a A = 0. To say it another way, were E a A = 0, Eq. (28) would constitute an infinite set of conservation laws, one for each choice of f (x). But there are no physical reasons for such conservation laws to exist, and we have no experimental indication that they do exist. We can only obviate them by admitting that E a A = 0. Now because in a generic configuration |φ| 2 cannot be a spacetime constant, we can only satisfy Eq. (25) by having E φ = 0. Of course by Eq. (20) it also follows that E φ * = 0. Thus we have again found E a A = E φ = E φ * = 0, i.e. all the equations of motion for the problem have been established without invoking the LAP.

IV. FERMION INTERACTING WITH NONABELIAN GAUGE FIELD
A. The gauge theory The standard model of particle physics abounds with examples of spin-1 2 fermions interacting via gauge fields. We thus turn attention to such a case, a spin-1 2 fermion in interaction with a non-Abelian gauge field. The action, again formulated in flat 4-D Minkowski spacetime, is given by [8] where D a ≡ ∂ a + ıgA aµ L µ is the gauge covariant derivative for the theory, and F µab = fields: A a ≡ L µ A µa and F ab ≡ L µ F µab . By analogy D a ψ = (∂ a + ıgA a )ψ. Since the L µ are independent it is easy to recover the individual A µa from the A a , etc.
The fermion fields are taken to transform as ψ → Uψ andψ →ψ U † . In addition, one prescribes the transformation law as a result of which the field matrix and the covariant derivative transform covariantly, namely F ab → UF ab U † and D a → UD a . The action (29) can now be rewritten as By making the above-mentioned transformations of F ab , D a , ψ andψ and invoking the cyclic invariance of the trace of a product of matrices it is immediate to see that S and its Lagrangian density are both unchanged, i.e. gauge invariant.
We may denote the variational derivatives with respect to the system's field variables by Thus to any increment of the fields corresponds the increment of the action with where ∇ b F ab µ ≡ ∂ b F ab µ + g C σνµ A νb F ab σ defines the gauge covariant divergence of a tensor. As before, we have dropped the boundary terms; this is natural when the boundary lies at infinity and A νb , ψ,ψ and their derivatives vanish asymptotically.

B. Scale and gauge symmetry lead to the gauge field equations
We now observe that the action (29) is also invariant under the scale transformation The metric is again regarded as unchanged. Under an infinitesimal scale transformation with ε = 1 + δε we have that δ ε A µa = −δε A µa , δ ε ψ = − 3 2 δε ψ, δ εψ = − 3 2 δεψ, δ ε m = −δε m and δ ε d 4 x → 4δε d 4 x. Hence Here we drop boundary terms under the conditions already mentioned. The above result can be be written as (E a A = L µ E a Aµ ) We already know that L is gauge invariant. Therefore, Tr (E a A A a ) d 4 x must itself be gauge invariant.

Now according to Eq. (36) E a
Aµ is the sum of the gauge covariant derivative of F ab µ and the quantity −gψγ a L µ ψ which is itself known to be gauge covariant [8]. This means means that under a gauge transformation E a A → UE a A U † . Employing Eq. (30) we see that This tells us immediately that Tr E a A U † (∂ a U) d 4 x = 0. Now to first order in the Λ µ s, U = 1 − ıL µ Λ µ + O(Λ 2 ). Therefore, because Tr(L ν L µ + L µ L ν ) = 2δ νµ we get, correct to Assuming that the Λ µ vanish asymptotically we get by Gauss theorem that Λ µ ∂ a E a Aµ d 4 x = 0. But since the Λ µ are all arbitrary functions, this tells us that for every µ, ∂ a E a Aµ = 0, or equivalently, ∂ a E a A = 0 all over spacetime. Of course this last result must be gauge invariant (we did not choose a specific gauge in deriving it). Thus for arbitrary U we should also have ∂ a (UE a A U † ) = 0. Substituting U here and retaining terms only up to O(Λ µ ) we obtain so that the term of O(Λ µ ) must vanish. Substituting for the commutator of the L µ and taking into account that the L µ are independent, we infer that for any τ Now since the Λ µ are arbitrary functions, the 4N quantities ∂ a Λ µ can be taken to be arbitrary and so independent of one another at any chosen event. But then the above mentioned linear combination of the 4N quantities E a Aν cannot vanish identically as required unless all the E a Aν themselves do so. This conclusion can be drawn separately for each τ . According to Eqs. (3) and (32) the conditions E a Aν = 0 we have just inferred without appealing to the LAP are precisely the gauge field equations for the system, to wit

C. Spinor field equation: Abelian case
We proceed to infer the field equations for the spinor fields. This turns out to be no easy task, so in this subsection we limit discussion to that of a single Abelian gauge field interacting with a singlet spin-1 2 field. In this case the role of L ν is played by unity, and the multiplet degenerates to a single 4-spinor (N = 1).
Consider then the (vanishing) increment (33) in the action (29) due to the infinitesimal local gauge transformation U = 1 − ıΛ + O(Λ 2 ). We substitute E a A = 0 from Eq. (44) to get But Λ is arbitrary, so we get at every spacetime point We can substitute in this Eqs. (34) and (35) to obtain the local conservation law of U(1) charge, e.g. electric charge: We stress that this result is in part off-shell since we have not yet identified the spinor field equations. We now proceed to that task, in which critical use will be made of Eq. (46).
We return to the (vanishing) increment, Eq. (33), of the action (29), but this time induced by the diffeomorphism x a → x a + χ a , where χ a is an infinitesimal but otherwise arbitrary spacetime dependent vector field. First the contribution from the increment in metric vanishes upon enforcement of the local conservation of energy-momentum. Next we take into account that E a Aµ = 0. Thus we get where the increments in the spinors are produced by Lie dragging them [9], namely, Integrating by parts in Eq. (48) and dropping boundary terms as done earlier we obtain Thus on account of the arbitrariness of χ b we infer that Taking the partial derivative ∂ a of both sides and exploiting the antisymmetry of σ ab we obtain the simplified relation This looks like a continuity equation; it is gauge covariant as can be verified by replacing ψ together with its conjugate version, and carrying out a local Abelian gauge transformation.
However, there is no room for a new conservation law such as Eq. (52). It does not correspond to charge conservation which would require a vector current formed without any derivatives of ψ. Neither does it correspond to energy-momentum conservation since the quantity whose divergence is zero here is a vector, not a symmetric tensor. The best guess is that the above result, far from being some serendipitous conservation law, is vacuous, i.e., Can we escape this conclusion? Several possibilities offer themselves. For example, V b could be proportional to a Killing vector, which would automatically nullify its divergence as required. Of course such situation would obtain only in the presence of a spacetime symmetry; for a generic field configuration there would be no Killing vector. Another possibility would be if all the components of V b were proportional to (−g) −1/2 . This would also nullify the divergence. However, in such case V b (−g) 1/2 would be a 4-vector with constant contravariant components; such a "constant" vector-a fixed spacetime direction-simply has no place in a generic solution. We must thus conclude that it is very hard to avoid Eq. (53).
These set of equations is somewhat reminiscent of Eq. (9) which we used to obtain the field equations for the scalar fields in Sec. II. The difference here is that we have only four equations (one for each of the four coordinates) constraining, so it seems, eight quantities in all, the components of E ψ and of Eψ. However, we receive assistance from Eq. (46). Suppose we multiply this last by an arbitrary complex vector t b (with no spinor aspects whatsoever), and add the result to Eq. (53) to get Now we take the Hermitian conjugate of this. In reworking the result we take into account that Hermitian conjugation inverts the order of all matrices and spinors besides conjugating each. We further take into account thatψ ≡ ψ † γ 0 and that γ 0 (γ a ) † γ 0 = γ a for all a (spatial γs are antihermitian). We thus get In Eqs. (54) and (55) we have, counting coordinate components, a total of eight linear equations in the eight components of the spinors E ψ and Eψ. In fact we can write the system where "T " signifies transpose of the relevant spinor. It is clear that the 8 × 8 matrix here does not have trivially identical rows or columns for a generic spinor field configuration. For one thing the t b is an arbitrary set of fields. And in generic spinor field configuration the relation betweenψ and ψ T will not be a simple one. Thus the determinant of the matrix can vanish only at isolated points or surfaces of some order. By continuity this allows us to conclude that in a generic field configuration E ψ = Eψ = 0 everywhere in spacetime, i.e.
that the spinor field equations are satisfied. Again, it is obvious that this conclusion did not entail use of the LAP.

D. Gauge symmetry and nonabelian charge conservation
Returning now to the nonabelian theory we reconsider the increment (33) in the action (29) due to the infinitesimal local gauge transformation U = 1−ıL µ Λ µ +O(Λ 2 ). Substituting which by virtue of the arbitrariness of the Λ µ says that We now substitute Eqs. (34) and (35) into Eq. (58). The mass terms cancel leaving us with Employing the group commutation law L ν L µ − L µ L ν = ıC λνµ L λ we have This is the well known local conservation law for the SU(n) charges (one for each generator L µ ) [7,8]. The form of Eq. (60) parallels that of Eq. (44), which informs us that the former is also gauge covariant.
As in the abelian case, the above derivation of the charge conservation laws is, at least partially, off-shell, i.e. we have not yet found the equations of motion for ψ andψ.

E. A novel partial symmetry
While not immediately related to our main line of reasoning, the following discussion turns up a novel symmetry in the system (29). We observe from Eq. (34) that the Dirac part of the action (including the gauge interaction term) from Eq. (29) can be written Consider now the unitary transformation ψ → Wψ andψ →ψ W † with where the ǫ ν are real constants and we have added to the generators L 0 = I, the unit matrix of the relevant rank. In the SU(n) case there will be n 2 different ǫ ν s. The gauge potentials A νa are to be regarded as entirely unchanged under the said transformation, which thus differs from a global nonabelian gauge transformation, c.f. Eq. (30). Now because W commutes with the γ a , it is immediate to see thatψEψ →ψEψ +gA µaψ γ a (L µ −W † L µ W)ψ.
Expanding W in a series in the ǫ µ and keeping up to first order terms, we get L µ −W † L µ W = −ıǫ ν [L ν , L µ ] = ǫ ν C λνµ L λ . Accordingly under the said transformations where use has been made of Eq. (60).
By Gauss' theorem the spacetime integral of ∂ a J a µ reduces to a boundary term. Consequently, in essence the total action (29) is invariant under the unitary transformation ψ → Wψ,ψ →ψ W † , A µa → A µa . By this we mean that the only change is addition of a "surface term" which leads, so we know from the standard approach, to no changes in the field equations. Unlike the usual symmetries, this one requires that we already impose the gauge field equations of motions, Eq. (44), which are the predecessor of the charge conservation laws, Eq. (60). It must be stressed, however, that no use has been made of the spinor equations of motions which have yet to be formally identified.
F. Spinor field equations: nonabelian case Still using the notation introduced in Sec. IV E, let us now combine all laws of the form (46) and (58) into the equation We recall that E ψ stands for a row comprising N adjoint 4-spinors E ψ (j) . Thus the first term in the equation above is the scalar product of this row with a column with N entries, each of which is a linear superposition of the Dirac 4-spinors ψ (j) that make up ψ (the L ν responsible for the superposition do not mix the individual components of each 4-spinor ψ (j) ).
The second term is identical to Eψ T (ǫ ν L ν TψT ) (again, the "T " here stands for transpose of the spinor rows or columns) whose structure is the same as that of the first term, except that the Dirac 4-spinors being superposed are now theψ (j)T . Of course the above establishes the field equations only for representations with limited dimensions. In SU(n), for example, the fundamental representation is of dimension n, and 2n is less than n 2 − 1 for all n ≥ 3. Thus for SU (3), SU(4), . . . the field equations in the fundamental representation of the fields can be obtained by the method just described.
For higher order representations all experience suggests that one obtains the correct field equations for the spinor fields by just replacing the fundamental representation's L ν by the corresponding matrices of the higher order representation, and introducing multiplets of the correct order. Thus the method here described should be generally applicable to all physical systems deriving from the action (29) and subject to many practically interesting symmetry groups. Again, it is clear that the LAP does not enter at any point.

V. SUMMARY
Traditionally for any classical field theory (defined by an action) the least action principle There are, of course, disadvantages to the latter method. The strategies for the solution vary from case to case, in contrast to the very standardized procedure for implementing the LAP. Further, the computations required by the symmetries-based procedure tend to be intricate. Thus, in practice, the latter method will not replace the LAP approach. However, provided it is a generally successful approach, the latter method shows that the LAP is not an essential part of a classical field theory. In our opinion this relieves the physics of a certain, almost metaphysical, element whose connection with experiment is tenuous at best.
For this alternative road map to the field equations to be generally applicable an obstacle must be surmounted. As well illustrated by the scalar fields example, and the spinor-gauge field one, the number of conservation laws may not suffice in every case to determine field equations for all field components. That is, overly complex physics, e.g. five interacting scalar fields in four dimensions, and on occasion very simple systems, e.g. SU(2) gauge theory with spinors, may not be covered by the scheme that sidesteps the use of the LAP.
This issue is still moot, and provides material for further research.