We discuss a relativistic particle affected simultaneously by two static potentials, a Lorentz scalar and the time component of a four-vector. We show that the scalar potential has effects on the motion of a particle that are distinctively different from those of a four-vector potential. With the aid of an example we show that, in both classical and quantum mechanics, there are two inequivalent but compatible criteria of nonrelativistic behavior: (mean) kinetic energy much smaller than rest energy and nonrelativistic potential energy independent of total energy. The second of these is the correct condition for use of nonrelativistic methods, but it is the first that is usually invoked to justify their use. The two criteria are equivalent only when the scalar potential has a (mean) value much less in magnitude than the particle’s rest energy. If this condition is not satisfied, nonrelativistic formalisms such as ordinary Newtonian mechanics and the Schrödinger equation may be inadequate even if v/c is small. On the other hand, it is possible that a large S may permit relativistic energy relations to be written in a nonrelativistic form even if v/c is not small. Such an influence of the size of the scalar potential on what is considered nonrelativistic behavior is unfamiliar and is rarely mentioned in textbooks and other publications. In our opinion, it should be better known.
We discuss a relativistic particle affected simultaneously by two static potentials, a Lorentz scalar and the time component of a four-vector. We show that the scalar potential has effects on the motion of a particle that are distinctively different from those of a four-vector potential. With the aid of an example we show that, in both classical and quantum mechanics, there are two inequivalent but compatible criteria of nonrelativistic behavior: (mean) kinetic energy much smaller than rest energy and nonrelativistic potential energy independent of total energy. The second of these is the correct condition for use of nonrelativistic methods, but it is the first that is usually invoked to justify their use. The two criteria are equivalent only when the scalar potential has a (mean) value much less in magnitude than the particle’s rest energy. If this condition is not satisfied, nonrelativistic formalisms such as ordinary Newtonian mechanics and the Schrödinger equation may be inadequate even if v/c is small. On the other hand, it is possible that a large S may permit relativistic energy relations to be written in a nonrelativistic form even if v/c is not small. Such an influence of the size of the scalar potential on what is considered nonrelativistic behavior is unfamiliar and is rarely mentioned in textbooks and other publications. In our opinion, it should be better known.
Classical mechanics of a free particle is based on Newtonian mechanics and Lagrangian/Hamilton theories. A single frame of reference is chosen (the lab) and particles with the same mass mo (for example) moving at different velocities are considered in the free particle scenario (i.e in the absence of force (potentials)). We argue that the above is identical to considering a single particle with mass mo at time=to and x=0 viewed by various frames moving at different constant velocities -v. From the point of view of each frame, the particle moves at a velocity v. Thus the Newtonian picture of an mo particle moving at different speeds may be considered as a problem of special relativity. We argue (as in (1)) that (x,t)=(0,t) transforms to (x’,t’) such that x’/t’=v. If one uses a metric (1 0) (first row) (0 -1) second row, one may derive special relativistic relationships. In particular, one may obtain the Lorentz invariant -Et + px if (p,E) and (x,t) transform in the same manner. We argue that the relationships obtained from special relativity pertain to classical mechanics and so one may mathematically obtain a function L(v) such that p=dL/dv and E=pv - L for a free particle. In other words, special relativity governs Lagrangian results in the free particle case and is not really needed (in this special case). More importantly, it is special relativity which links E with t and p with x. As noted above, if one considers Netwonian mechanics in a single frame, the notion of special relativity does not arise, but the scaling of time by E and of x by p does i.e. A= -Et+px is the classical action both relativistically and nonrelativistically. It is just not written in this way usually (i.e. A = -mo t sqrt(1-vv) (c=1) relatiivstically and A =t .5mo vv nonrelativistically) We argued in (2) that scaling must manifest itself physically and that it accounts for the physical wavelength in quantum mechanics. We argue that the quantum mechanics of a free particle ultimately follows from special relativity. As a result, ideas of quantum mechanics would not have appeared in Newtonian mechanics nor Lagrangian or Hamiltonian theories. When special relativity was developed it focused on transformations related to different frames, but these are equivalent to describing one frame with many different velocities (same mo). Thus the physics in one frame is governed by special relativity even if it is nonrelativistic. We argue that the notion of the wavelength which appears even in the Schrodinger formalism follows from scaling of x by p which ultimately originates from special relativity. In other words, both the Lagrangian formalism and quantum mechanics follow from special relativity in the free particle case. We argue that free particle quantum mechanics is the starting point of overall quantum mechanics.
In (1) a Galilean transformation x’ = x-vt and t’=t is applied to the Schrodinger equation id/dt (partial) W(x,t) = -1/2m d/dx d/dx W(x,t) for a free particle, where W(x,t) is the wavefunction, and it is argued that W’(x’,t’) does not equal W(x(x’t,t’),t(x’,t’)). Instead it is shown that W’(x’,t’) = W(x,t) exp( -i/bar [mvx’+.5mvvt’] with x=x’+vt’ and t=t’. The author’s of (1) suggest this is due to remnants of special relativity creeping into the nonrelativistic situation. They then argue that this issue is already found in nonrelativistic classical mechanics, in particular in the Hamiltonian-Jacobi equation H(x, dS/dx) + dS/dt = 0 where S is the free particle action= t *Lagrangian and H is the Hamiltonian i.e 1/2m dS/dx dS/dx. They show that S’(x’,t’) = S(x,t) - (mvx’ +.5mvvt’) and also argue that this is due to a remnant of special relativity creeping into nonrelativistic classical mechanics. In this note, we consider the free particle action S= -Et+px nonrelativistically. (This form also holds for the relativistic free particle). It may be seen that such a solution satisfies the Hamilton-Jacobi equation, but the catch seems to be that one must transform both (x,t) and (p,E) to find a solution in a moving frame. The Galilean transformation leads to two different transformations, one for (x,t) and another for (p,E). A Lorentz transformation on the other hand treats (x,t) and (p,E) in the same way and -Et+px behaves as a dot product with a (1,-1) type metric. We argue that given the physical example of a particle with rest mass mo at x=0 at t=to, a moving frame transforms this into a new energy (including mo and kinetic energy), a p and x’,t’ such that x’/t’=v where -v is the velocity of the frame. One may suggest a more general transformation than the Galilean i.e. replace t’=t with t’=g(v) t. This then leads to -Et+px written in terms of E’,t’, p’,x’ equalling -E’t’ + p’x’ if g(v) = 1/sqrt(1-vv/bb) where b is a constant to make v/b dimensionless. Applying the transformation, which we assume holds for both (x,t) and (p,E) to light yields b=c. One may then show that -Et+px, written in terms of E’,t’,p’,x’, yields -E’t’ + p’x’,
The classical action (relativistic or nonrelativistic) may be written with v=x/t and x and t varied independently to yield dA/dx partial = p and dA/dt partial = -E. Thus A = -Et+px is a solution. This expression also appears in classical wave forms and in the electromagnetic description of a photon (from Maxwell’s laws). -Et+px is a Lorentz invariant and so is directly linked to special relativity and different frames moving with constant velocity. In (1) we argued that x and t should be treated as independent in A. This suggests that p is a scaling in space and E in time. Thus there should be physical lengths proportional to 1/p and physical time intervals proportional to 1/E. Special relativity considers different frames (moving with constant velocity) and 4-vectors so it seems to be quite different from a scenario with t and x separated and scaling of x by p. We argue that although -Et+px is a Lorentz invariant and as such may be used to find E(v) and p(v), quantum mechanics is usually performed in a special frame, namey one in which V(x) or V(x,v) the potential is at rest. (For electromagnetism one treats (A, phi) as a 4-vector, but this is not the norm.) As a result, energy En found in a rest frame of the potential also represents “rest energy” so En is binding energy which is added to rest mass energy. In other words motion in the rest frame of a potential creates rest energy = rest mass. We argue that V(x) may be written as a sum over k Vk exp(ikx) and so no time related terms appear, only momentum linked ones. Thus momentum is the key feature for the interaction, even though velocity and time are still in the picture (just sometimes not directly in the interaction scheme). For example, a two slit interference or single slit diffraction does not depend on time or speed although these are certainly in the problem. For a bound state v and t are also in the problem as one calculates average kinetic energy using pp/2m = .5 p v which is rate dependent, but there is no acceleration which is a time-dependent phenomenon. Rather one deals with an ensemble of exp(ipx)’s because these interact with the potential in a wavelike manner to create the effect of acceleration. exp(ipx) a momentum wave linked to the scaling of x by p in px is not only involved in an interaction with V(x), but represents extra information from the classical mechanical picture. exp(ipx) for a known rest mass mo means one knows v=x/t=dx/dt, yet there is an extra piece of information namely the phase px which is involved in even “free” interactions such as two idendical particle with different phases arriving at the same x at the same t or the average kinetic energy in a bound state. Velocity v is converted into a momentum expression through p/mo in the norelativistic case and so exp(ipx) governs the whole scenario. Acceleration then becomes a mathematical abstraction of the motion of vrms(x) for any quantum bound energy level with the real physics associated with exp(ipx) being ignored.
It is generally acknowledged that neither the Klein-Gordon equation nor the Dirac Hamiltonian can produce sound solitary-particle relativistic quantum mechanics due to the ill effects of their negative-energy solutions; instead their field-quantized wavefunctions are reinterpreted as dealing with particle and antiparticle simultaneously--despite the clear physical distinguishability of antiparticle from particle and the empirically known slight breaking of the underlying CP invariance. The natural square-root Hamiltonian of the free relativistic solitary particle is iterated to obtain the Klein-Gordon equation and linearized to obtain the Dirac Hamiltonian, steps that have calculational but not physical motivation, and which generate the above-mentioned problematic negative-energy solutions as extraneous artifacts. Since the natural square root Hamiltonian for the free relativistic solitary particle contrariwise produces physically unexceptionable quantum mechanics, this article focuses on extending that Hamiltonian to describe a solitary particle (of either spin 0 or spin one-half) in relativistic interaction with an external electromagnetic field. That is achieved by use of Lorentz-covariant solitary-particle four momentum techniques together with the assumption that well-known nonrelativistic dynamics applies in the particle's rest frame. Lorentz-invariant solitary particle actions, whose formal Hamiltonization is an equivalent alternative approach, are as well explicitly displayed. It is proposed that two separate solitary-particle wavefunctions, one for a particle and the other for its antiparticle, be independently quantized in lieu of "reinterpreting" negative energy solutions--which indeed don't even afflict proper solitary particles.
In classical physics it is well known that a stationary solution of the classical action Integral dt L ,where L is the Lagrangian, leads to a differential equation whose solution x(t) describes classical mechanical motion. In previous notes we have argued that two free particle flow/flux equations d/dx partial A(x,t) = p and d/dt partial A(x,t) = -E lead to A(x,t) being the classical free particle action with a solution: A(x,t)= -Et+px. In this note, we consider two stationary solutions i.e. dA=0 which differ from the classical approach. One is for a particle with rest mass for which we assume p=E(v)v. The stationary behaviour leads to the form (Lorentz matrix) of special relativity and also to relativistic/nonrelativistic quantum mechanics for a free particle. The second dA=0 solution leads to a zero rest mass particle for which the Lorentz transformation of the first dA=0 solution also holds because (x,t) transforms in the same manner for both zero and nonzero rest masses. We argue that dA=0 i.e. the action is stationary because A=-Et+px is a Lorentz invariant i.e. a dot product.
The formal mathematical structure of the truly Hamiltonian formulations of nonrelativistic and relativistic quantum mechanics is critically examined for both spin 0 and spin 12 particles with nonzero mass. It is shown that the relativistic quantum mechanics can, in principle, at least be fitted into a truly covariant Hamiltonian procedure where the Hamiltonian is a Lorentz-invariant world scalar (and is the negative of 12 the rest energy of the particle, regardless of its spin). In particular, a truly covariant Hamiltonian theory of the electron is presented. It will be seen that the Lorentz-invariant Dirac theory of the electron does not fall within the framework of a truly covariant Hamiltonian procedure. Further, the dynamical variables associated with the translational degree of freedom of the particle (such as the position, the canonical momentum, and the velocity four-vectors of the relativistic quantum mechanics) correspond precisely to the same variables of the truly covariant formulation of relativistic classical mechanics. The mathematical structure of relativistic quantum mechanics is seen to be rather similar to that of nonrelativistic quantum mechanics for both spin 0 and spin 12 particles with nonzero mass.