The progress of physics in the twentieth century is deeply associated with the discovery of the theory of relativity and the quantum theory. Relativistic and quantum theories form the basis of modern understanding of the surrounding world. The dependence of space distances between events on the frame of reference is usually called the relativistic length contraction or Lorentz-Fitzgerald contraction of a moving body. In transition from one reference frame to another by Galilean transformations, the time interval between any two events remains the same as well as the distance between the space points where the events occurred. One photon of a sufficient energy can generate an electron-positron pair only in the presence of some other particle that would provide the fulfillment of the law of momentum conservation. In the latter case the energy and the momentum conservation laws for an act of a photon emission by a free moving atom.
The progress of physics in the twentieth century is deeply associated with the discovery of the theory of relativity and the quantum theory. Relativistic and quantum theories form the basis of modern understanding of the surrounding world. The dependence of space distances between events on the frame of reference is usually called the relativistic length contraction or Lorentz-Fitzgerald contraction of a moving body. In transition from one reference frame to another by Galilean transformations, the time interval between any two events remains the same as well as the distance between the space points where the events occurred. One photon of a sufficient energy can generate an electron-positron pair only in the presence of some other particle that would provide the fulfillment of the law of momentum conservation. In the latter case the energy and the momentum conservation laws for an act of a photon emission by a free moving atom.
Special relativity seems to be associated with linking physical relations as seen in one frame with those as seen in a frame moving at a speed relative to such a frame. A constant speed is often used for the moving frame. This suggests that classically there is a special frame for a particle with rest mass called a rest frame. In such a case, one simply introduces a parameter for rest mass and does not worry about the complicated motion and interactions which actually create the rest mass. It seems that even quantum mechanics, which considers a particle to be moving as one does not know its location, one uses the idea of a potential V(x) which in principle is associated with a very large rest mass, presumably at rest as no variables of such a rest mass appear in the Klein-Gordon or Dirac equations. It seems that one uses a preferred frame when solving relativistic quantum mechanical equations. Furthermore, given V(x) and a bound state with exp(-i E bound t), E applies to the overall bound system, not to a possible exp(ipx) constituent. The special relativistic invariant equation seems to exist at the system level, like with the rest mass and one may apparently calculate E in the special rest frame of the large mass and hence find a rest mass of the bound system, i..e large rest mass*cc + mo cc + E, mo is the rest mass of the single bound particle. As a result of the use of the special rest mass frame for the V(x) producing large rest mass, it seems that one does not have to worry about special relativistic constraints on the interaction with V(x), as only one frame is involved. In other words, a bound calculation is about finding a new rest mass for an overall new system. Then this system may be used in Lorentz boosts. One simplifies by working in the rest frame of the mass creating V(x). Perhaps surprisingly then, we note that exp(-iEt+ipx) for a free particle is Lorentz invariant and in fact must be Lorentz invariant for the following reason. Imagine that one wishes to create a probability for free particles which is linked with conservation of momentum. Any p must have the same real value weight because for free particles there is no bias for one p or the other. There is, however, a bias for a p1 and p2. It should have the same probability as a p3 and p4, if p1+p2 = p3+p4. (We use one dimension for simplicity.) As a result, exp(ip) seems to be a probability which respects these conditions. The problem is that one may have p=m1v1=m2v2 hence (E1,p) and (E2,p). With respect to momentum, it is irrelevant if one particle has E1 and the other E2, but not if one does a Lorentz boost. Then, in one case: p→ g(v) p + g(v) v E1 and in the second, p→ g(v) p + g(v)vE2. Here g(v) =1/sqrt(1-vv/cc). In other words, p is the same for each particle in one frame and not in the other which violates conservation of momentum because in frame 1 both particles can complete the same conservation of momentum equation, but in the second frame this is not the case, because each has a different momentum. Thus, exp(ip) cannot hold. One requires a linear form in p which is Lorentz invariant, i.e. exp(-iEt+ipx), we argue (and have done so in previous notes). We see now that momentum probability is linked with space in exp(ipx). The idea seems to be that for free colliding particles there is no preferred frame and the probabilities used must reflect this Lorentz invariance. As a result, we argue that a free particle probability in E,p arises because of Lorentz invariance, but then apply it to a V(x) problem and no longer worry about Lorentz invariance at the interaction level. V(x) with no associated rest mass variables assumes a rest frame for the V(x) producing mass. A Lorentz boost which then invalidates exp(ip), would only apply to the total rest mass of the bound particle system which has no overall momentum. It seems that one does apply the Lorentz transformation to the interaction of exp(ipx)’s with V(x). One is only interested in the overall bound system as a single entity for special relativity. As a result, one may only impose momentum considerations as V(x) is a spatial function and one must link it with exp(ipx). In fact, one may write V(x) = Sum over k Vk exp(ikx) to associate it with momentum hits linked with Vk energies, which already moves away from the idea of special relativity as Vk and k are not linked as a 4-vector. The idea of exp(ip1x)exp(i k1x) = exp(ip2x)exp(ik2x), however, seems to still hold (as it does for free particles), linking these two values (i.e. Vk1 and Vk2) with (p1+k)(p1+k) exp(i (p1+k)x). Lorentz invariance for a particle with rest mass seems to be linked with the idea that one ultimately has a rest frame for the particle. One then establishes a relation between the rest mass in one frame and its new E and p as seen by another frame. One does not, however, need to impose Lorentz invariance conditions on the interaction which creates the rest mass it seems. For example, exp(-iEt+ipx) is the Lorentz form used to describe a free particle, but exp(ipx) is the only portion used to describe its interaction with V(x) in the rest frame of the large mass creating V(x). Furthermore, exp(-iEt+ipx) ensures conservation of energy and p for a free particle, but in an interaction case, that does not happen as the potential may absorb energy.
The early twentieth century saw two major revolutions in the way physicists understand the world. One was quantum mechanics itself and the other was the theory of relativity. Important results also emerged when these two ideas were brought together: some of these have been referred to in earlier chapters-in particular the fact that fundamental particles such as electrons have intrinsic angular momentum (spin) was stated to be a relativistic effect. This chapter explores the relationship between relativity and quantum mechanics more deeply A full understanding of relativistic quantum mechanics is well outside the scope of this book, but many of the important results can be understood at this level and these will be discussed in this chapter. After a short summary of the main results of special relativity, we show how combining this with the time-dependent Schro¨dinger equation leads to a new wave equation known as the Dirac equation. We show how the Dirac equation requires particles such as electrons to have intrinsic angular momentum (spin) and we explore some of its other consequences. The chapter concludes with an outline of some more advanced ideas known as quantum field theory.
Abstract The steady development of the quantum theory that has taken place during the present century was made possible only by continual reference to the Correspondence Principle of Bohr, according to which, classical theory can give valuable information about quantum phenomena in spite of the essential differences in the fundamental ideas of the two theories. A masterful advance was made by Heisenberg in 1925, who showed how equations of classical physics could be taken over in a formal way and made to apply to quantities of importance in quantum theory, thereby establishing the Correspondence Principle on a quantitative basis and laying the foundations of the new Quantum Mechanics. Heisenberg’s scheme was found to fit wonderfully well with the Hamiltonian theory of classical mechanics and enabled one to apply to quantum theory all the information that classical theory supplies, in so far as this information is consistent with the Hamiltonian form. Thus one was able to build up a satisfactory quantum mechanics for dealing with any dynamical system composed of interacting particles, provided the interaction could be expressed by means of an energy term to be added to the Hamiltonian function. This does not exhaust the sphere of usefulness of the classical theory. Classical electrodynamics, in its accurate (restricted) relativistic form, teaches us that the idea of an interaction energy between particles is only an approximation and should be replaced by the idea of each particle emitting waves which travel outward with a finite velocity and influence the other particles in passing over them. We must find a way of taking over this new information into the quantum theory and must set up a relativistic quantum mechanics, before we can dispense with the Correspondence Principle.
Ψ∂ (4.1) The harmonic trial solution of this equation with respect to a fixed inertial frame ( is given by )tx, ( ) ( ) ( )ctxikectxktx −Ψ=Ψ−Ψ=Ψ 00 orcos, In another inertial frame( ),, tx ′′ considered to be in uniform translational motion with velocity as described by the Galilean transformation (2.13), with respect to the fixed frame, the wave is represented by a wave function ,v ( ) ( ), related to ,x t x′ ′Ψ Ψ t as 2 2i.e., x t x x x t x x x x ′ ′∂Ψ ∂Ψ ∂ ∂Ψ ∂ ∂Ψ ∂ Ψ ∂ Ψ= + = =′ ′ ′ ′∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ and 2 2i.e., 2 x t v v v t x t t t x t x tt x ′ ′∂Ψ ∂Ψ ∂ ∂Ψ ∂ ∂Ψ ∂Ψ ∂ Ψ ∂ Ψ ∂ Ψ ∂ Ψ= + = − + = − +′ ′ ′ ′ ′ ′ ′∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ 2t∂ ∂ ∂ where use has been made of known properties of partial derivatives. It follows that obeys the following wave equation, with respect to the moving inertial frame ( tx ′′Ψ , ) ( ) tx v tx vc ′∂′∂ Ψ∂−′∂ Ψ∂=′∂ Ψ∂− 2 22 2 (4.2) which is obviously different from the wave equation (4.1) that holds in the fixed inertial frame. In other words the electromagnetic wave equation is not invariant with respect to a uniform translation of the inertial frame, using a Galilean transformation. Inserting into Eq.(4.2) harmonic solution written as we obtain 2 2 2 2 2 2( ) ( ) 2 ( ) i.e., orc v k k w v k w w v c w c− − = − − = − ± = ± v (4.3) Hence Galileo's transformation predicts that the velocity of light should be different from c, if measured with respect to a moving inertial frame. We therefore have a choice to make before we can fully understand the electromagnetic phenomena. We can either accept Galileo's principle of relativity which states that all inertial frames are equivalent or the Galilean transformation (2.13). We can sacrifice Galileo's principle of relativity by saying that there is a privileged inertial frame, called the ether, in which it is supposed that the electromagnetic wave velocity is c as given by Maxwell's equations, although they made no essential reference to the ether. A long series of investigations has been carried out, in order to detect some effect of the motion of the earth through the ether, always with negative results. All of the experiments (the best known being that of Michelson-Morley) have indicated that the speed of light is always the same in all directions and is independent of the relative motion of observer, source and transmitting medium. Einstein has reaffirmed Galileo's principle of relativity on the equivalence of all the inertial frames and, in addition, has assumed as a basic postulate the experimental fact that the speed of light is always constant. As a consequence, the Galilean transformation had to be replaced by another, in order to preserve the same speed of light in all systems. Einstein's basic assumptions are known as the postulates of special relativity: 1. The laws of physics take the same form in all inertial systems 2. The speed of light in empty space is a universal constant c and is the same for all observers. It is independent of the state of motion of the emitting body Special relativity is only concerned with transformations between inertial frames, as indicated by the restriction contained in the first postulate. General relativity covers reference frames in arbitrary relative motion, and is based on the principle of equivalence, confirmed experimentally, which unifies gravitational and inertial forces. The first of Einstein's postulates implies that it is only possible, by means of physical measurements, to demonstrate that two gravity-free reference frames are moving relative to each other. There is no physical basis to infer that any such frame is intrinsically at rest or in uniform motion. The notion of absolute rest is not consistent with the first postulate. The second postulate is in contradiction with the Galilean transformation which would preserve the form of Newton's laws of motion, as illustrated by Eq.(4.3). Therefore, we must find a transformation between two frames moving with uniform relative motion, which is consistent with the second postulate. We must then generalize the laws of Newtonian mechanics in a form which is invariant under that transformation, as required by the first postulate. A fundamental requirement of special relativity is that the laws of Newtonian mechanics are valid in the low speed limit and must be obtained as limiting cases of relativistic formulae for .cv << This assumption plays, for special relativity, a role analogous to that of Bohr's correspondence principle in quantum mechanics (see Chapter 28). 4.2. THE LORENTZ TRANSFORMATION Einstein's procedure is to locate an event in a reference frame by its spatial Cartesian coordinates x, y and z and a temporal coordinate t. Consider two frames S and S' moving with uniform relative motion, such that the Cartesian axis system in coincides with that in S, Oxyz, at time ,S ′ zyxO ′′′′ .0==′ tt As illustrated in Figure 4.1, the x and axes are parallel to the direction of relative motion: in moves in the x direction with velocity v, and in x′ , the frameS ′S S, the frameS ′ moves with velocity in the direction. A pulse of light emitted from O at time v− x′ 0=t spreads as a spherical wave travelling with speed c, so that the equation of the wave front in S, at a time t will be (4.4) 22222 tczyx =++ 0 0' z' y' x,x' z y Figure 4.1. Cartesian axes of two inertial frames in uniform relative motion The second postulate requires that the wave front in S ′ can be represented by (4.5) 22222 tczyx ′=′+′+′ In other words an observer in also sees a spherical wave travelling from its origin if not, any change of the wave front could be used to infer that the system is moving uniformly instead of being at rest. The invariance of the expression S ′ ,O′ S ′ (4.6) 2222222222 tczyxtczyx ′−′+′+′=−++ is a basic requirement that the transformation between the two systems must satisfy. The Galilean transformation (2.13), which in our case reads ttzzyyvtxx =′=′=′−=′ ,,, is not consistent with the invariance property (4.6), as it gives xvttvtczyxtczyx 2222222222222 −+−++=′−′+′+′ The first postulate brings some simplification to the possible form of the desired transformation. Since a uniform rectilinear motion in S must go over into a uniform rectilinear motion in S', the transformation must be linear. A nonlinear transformation would predict acceleration in S' even if the velocity were constant in S. The choice of the origins in space and time has avoided constant terms, and the transformation will be homogenous. The directions perpendicular to the relative motion, which are effectively at rest, must be left unchanged by the transformation. We then take the general form BtAxx +=′ yy =′ (4.7) zz =′ FtDxt +=′ Further simplification comes from the argument that the origin O', which has the coordinate moves along the x axis with velocity ,0=′x ,/ vdtdx = that is ABvBtAx /or0 −=+= while the origin O, with coordinate ,0=x moves along x-axis with velocity Recalling that when , we can write .'/' vdtdx −= 0=x 0=t 00 ort D B Bx A B t v F F F ′ − ⋅′ ′= ⋅ + = − = so that Inserting now Eqs.(4.7) into Eq.(4.5), we obtain the equation .FA = ( ) ( ) 02 22222222222222222 =−+⎟⎟⎠ ⎞ ⎜⎜⎝ ⎛ −−++−=′−′+′+′ DcBAxt c BAtczyxDcAtczyx The identity (4.6) requires that 2222 =−=−=− DcB c BADcA Since ,B vA= we obtain 2222 /1 /, /1 , /1 cv cvD cv vB cv A − −= − −= − = (4.8) so that Eqs.(4.7) now read 22 /1 cv vtxx − −=′ yy =′ (4.9) zz =′ /1 / cv cvxtt − −=′ The four equations (4.9) are known as the Lorentz transformation, which in this case applies to the relative motion of inertial frames along the x-axis. It is a straightforward matter to show that Eqs.(4.9) are consistent with the invariance relation (4.6), which is thus a sufficient as well as a necessary condition for the Lorentz transformation. With the usual abbreviations ( )1 /1 1, 222 ≥ − = − == γ β γβ cvc v (4.10) the Lorentz transformation simplifies to ( ) ( )x x ct x x ctγ β γ β′ ′= − = + ′ a) yyyy ′==′ or b) (4.11) zzzz ′==′ c) ⎟⎠ ⎞⎜⎝ ⎛ ′+′=⎟⎠ ⎞⎜⎝ ⎛ −=′ x c ttx c tt βγβγ d) The right hand side shows the inverse transformations where the primed and unprimed quantities are interchanged and the sign of the relative velocity is reversed, because its direction is the only difference between the systems S and S'. In the low velocity limit the Lorentz transformation reduces to the Galilean transformation since the only difference consists in the presence of the factor γ and the term ,1< that, in Eq.(4.11a), γ indicates a change of measuring stick length and in Eq.(4.11d) a change in the clock rate in the Lorentz transformation. The term ( )xc/β− in Eq.(4.11d) shows that the time is dependent on the position of a given event. Therefore, the immediate consequences of the Lorentz transformation require the revision of the usual concepts of simultaneity, time and length. Simultaneity Consider two events occuring at the same time ttt == 21 at two different positions and in the unprimed system S. According to Eqs.(4.11) the events are recorded in the primed system S' at different times 1x 2x / /, 1 1 t x c t xt t cβ ββ β − −′ ′= = − − so that the apparent time interval in S' is given by ( ) 0 1 / 2 21 12 ≠− −=′−′=′∆ β β cxx ttt (4.12) and hence, it is a function of the relative velocity v. The events will not appear as simultaneous to observers in S'. We are then forced to abandon the intuitive idea of absolute simultaneity, which is based on our experience with objects of low velocity, and accept the relativity of simultaneity, which depends on the coordinate system and applies in the domain of high velocities. Length contraction Consider a rod at rest in the primed system S' lying along the x-axis and having the length 12 xx ′−′=λ given by the distance between its ends 21 and xx ′′ measured at the same instant of time t'. The length in an inertial frame in which the rod is at rest is called its proper length. To define length in a frame S in which the rod is moving, an observer must locate the position of both end points, 21 and xx ′′ , at the same time From the inverse equations (4.11) we obtain 1 2 .t t t= = 1 1 2 1 x ct x ct x xx x x xβ ββ β β − −′ ′ ′ ′= = − = − − − − so that the apparent length 12 xxl −= is 21 βλ −=l (4.13) This equation is known as the Lorentz contraction and states that a rod parallel to the direction of its motion is shortened by the factor .1 2β− It results from the fact that simultaneous events in the rest frame of one rod are not simultaneous in the rest frame of the other. A rod transverse to its direction of motion is unchanged in length. The proper length λ is independent of the orientation of the rod, as a manifestation of the isotropy of inertial rest frame S'. If the orientation of a rod is neither longitudinal nor transverse to the direction of relative motion, both the length and the inclination are changed in the laboratory frame S, as an effect of a contraction in the direction of the relative motion with no change of transverse dimensions.
One of the concepts of Relativity theory that challenges conventional intuition the most is time dilation and length contraction. Usual approaches for describing relativistic effects in quantum systems merely postulate the consequences of these effects as physical constraints. Here, we propose to rebuild Special Relativity from quantum mechanical considerations. This is done by dropping one of its fundamental postulates: the universality of the speed of light. Lorentz transformations are obtained by a quantum mechanism. We use the fact that quantum states depend on the Galilean reference frame where they are defined. In other words, quantum states outside an observer's Galilean reference frame are distorted. Then, we show in a theorem the existence of time-dependent observables that are sensible to this distortion in such a way that their expectation value is a Lorentz-covariant function of time. We then postulate this mechanism as the source of the phenomena of Special Relativity. As a corollary of the main theorem, we show the existence of a Lorentz covariant momentum and mass operators which yield the relativistic momentum and mass. In this theory, the fundamental limit of the speed of light imposes a transparency condition for faster-than-light particles: they are allowed but they are not observable. The transparency effect could explain dark matter in a more general theory following this quantum formalism.
Unlike special relativity, quantum mechanics was not the inspired product of one mind. It developed, in fits and starts, over a period of 25 years (1900–1925), and many cooks stirred the broth. Even now, a century later, quantum mechanics raises profound conceptual questions. Every competent physicist can “do” quantum mechanics, and the predictions it makes are in spectacularly good agreement with experimental results; there can be little doubt that quantum mechanics is “right.” On the other hand, as Richard Feynman once remarked, “nobody understands quantum mechanics.” I'll tell the story in two parts. First the history (how the essential pieces of the puzzle were assembled), and then the implications (what the finished picture has to say about our world). Photons Like relativity, quantum mechanics began with the study of light. In both cases, this historical association is misleading. Relativity, really, has nothing to do with light – to be sure, it involves that magic speed limit, c , and of course the most familiar thing that travels at speed c is light. So light is a convenient vehicle for introducing the subject. But it is perfectly possible to imagine a universe in which there is no such thing as light, and yet relativity might still be valid. Quantum mechanics, too, starts with light – in this case the “quantum” of light (the photon). But, as it turns out, this is just one rather specialized example of much more general principles – in fact, a rather bad example, because photons are by their nature relativistic (they travel at the speed of light), and it would be more natural to begin with the quantum mechanics of slow-moving objects, in analogy with Newtonian mechanics.