It is demonstrated that a modification of the class of permissible variations in the Schwinger variational principle permits one to include spin variables in the general Lagrangian formalism in the nonrelativistic as well as relativistic cases. (auth)
Abstract The classical and quantum theories of dynamical systems whose Lagrangians are linear in the co-ordinate derivatives are studied, with a view to clarifying a number of points in relation to Schwinger’s quantum-mechanical variational principle. In particular, linear variations of the dynamical variables are considered, and it is shown that they may be consistently included in Schwinger’s formalism. It is also found to be possible to obtain Green’s generalized commutation rules by excluding c number variations from the theory, for a suitably restricted class of Lagrangian.
Components of a field function are functions of coordinates and time and satisfy some definite differential equations, which are called field equations. Field equations can be given a very general variational or Lagrangian form, if one accepts, that a field as the dynamic system must be characterized by a definite Lagrangian function. For a mechanical system this function is written down as the sum over all the material points of the system. For a continuous system like a wave field, Lagrangian function is expressed by the space integral from Lagrangian function density: L(t) = ∫ Ld3x. (4.1) However, the variational principle deals not with the Lagrangian function but with an action S obtained from it by means of integration over x0 = ct S = c ∫ L(t)dt. (4.2) Therefore, noncovariant expression (4.1) in Lagrangian formalism is in fact intermediate and it is sufficient to consider the Lagrangian function density L, which depends on the four space-time coordinates x in an implicit way. Hereinafter, we shall simply call L by the Lagrangian. Analogously, the Hamiltonian function density H will be named the Hamiltonian. The Lagrangian depends on components of a field wavefunction ψk and on their first derivatives with respect to the space-time coordinates ∂ψk/∂x ν . In so doing we assign the meaning of generalized field coordinates to ψk, and the meaning of generalized field velocities to ∂ψk/∂x ν . The demand of lacking derivatives with respect to field functions above the first order in the Lagrangian is caused by the following reason. Dynamics of all known quantum fields is described either by equations of the first or the second order. On the one hand, these equations follow from the principle of a least action: δS = 0, (4.3) and under action variation the order of derivatives, entering L, is raised by one. The Lagrangian must be a relativistically covariant quantity, namely, a relativistic scalar L′(x′) = L(x). Thanks to the invariance under the space-time translations, we conclude that the Lagrangian must not manifestly contain any space-time coordinates x. Since dynamical observables (four-dimensional vector of the energy-momentum, the moment of momentum tensor, and so on) are real quantities, then the Lagrangian must be also a real quantity. We shall also impose the locality condition on the Lagrangian, that is, we assume that the Lagrangian depends only on the state of fields in the infinitely small vicinity of the point Particles, Fields, and Quantum x. So: L(x) = L(ψk, ∂ψk ∂xν ) = L (ψk(x), ψk;ν (x)) . General variation of an action, related with varying both field wave functions and of integration domain boundaries, is equal to: δS = δ ∫ L(x)dx = ∫ L(x′)dx′ − ∫ L(x)dx, (4.4) where dx = dx0dr. For L′(x′) we have the expression: L′(x′) = L (ψ′k(x′), ψ′k;ν(x′)) = L(x) + δL(x) = L(x) + δL(x) + dLdxλ δxλ, (4.5) In Eq. (4.5) we denote a form variation of corresponding quantities by the symbol δ. This way δL(x) represents varying the Lagrangian only on the account of the variation of field functions and field function derivatives: δL(x) = ∂L ∂ψk δψk + ∂L ∂ψk;ν δψk;ν , (4.6) and δψk(x) does a variation of a field function form: δψk(x) = ψ ′ k(x) − ψk(x).
Date on paper should be 2022 not 2021. In a previous note, we argued that for a free particle there exists a function A(x,t) such that d/dx (partial) A(x,t) = p ((1a)) and d/dt (partial) A(x,t) = -E ((1b)). In particular, we used p=E(v) v for the relativistic case and p=mov. We showed that A(x,t) is the classical action for a free particle Integral dt L = T L with v=x/t before the derivatives are taken. After, x/t is set to v. As a result, there is a nonrelativistic A(x,t)= T .5mvv and a relativistic one A= -mo sqrt(1-vv) (c=1). We argued that ((1a)) and ((1b)) are in the form of flow equations. Energy E flows through space out of a dx region (i.e there is an energy flux Ev) so if x is held constant, the change in time is -E. We also noted that one may obtain both the Schrodinger and Klein-Gordon equations from ((1a)) and ((1b)) (see reference in (1)). As a result, A(x,t) is linked to both the nonrelativistic and relativistic free particle actions, yields free particle quantum mechanical equations and may also be used to obtain the form of special relativity (as discussed in the reference in (1)). In this note, we wish to consider spin which is not present in ((1a)) and ((1b)). For both relativistic and nonrelativistic classical dynamics which are linked to A(x,t) one deals with numbers (scalars) and so no notion of spin arises. In obtaining both the relativistic Klein-Gordon equation and nonrelativistic Schrodinger equation from ((1a)) and ((1b)), one creates an eigenvalue equation i.e. id/dt exp(i Action) = E exp(iAction) ((2a)) and -id/dx exp(i Action) = p exp(i Action) ((2b)) which only hold for the free particle. One may then use E and p in expressions EE = pp + momo (where p=Ev= dAction/dx partial) and E = pp/2m + V(x) where p=mov=dAction/dx (partial). For the free particle, one has ((2a)) and ((2b)) without establishing the Klein-Gordon equation. Given that p=Ev is a vector, one may try to write ((2b)) as a vector, but given that one has operators acting on exp(i Action) one does not use a basis vector ei (ei dot ei=1), but rather a “basis” operator or a spin matrix. Thus, we argue that spin emerges in this way without having to linearize the Klein-Gordon equation. Interestingly, this approach only holds for fermions. For the photon, which is a boson, a Dirac-like equation may be obtained directly from a d/dt (partial) EL EL + BB + b grad dot (El x B) = 0 as shown in (2) (with a,b being constants). This again follows from a flow scheme.
The classical and quantum theories of dynamical systems whose Lagrangians are linear in the co-ordinate derivatives are studied, with a view to clarifying a n-uiber of points in relation to Schwinger's quantum-mechanical variational principle. In particular, linear variations of the dynamical variables are considered, and it is shown that they may be consistently included in Schwinger's formalism. It is also found to be possible to obtain Green's generalized commutation rules by excluding c number variations from the theory, for a suitably restricted class of Lagrangian.