Let us start with Figure 1. During the early years of the 20-th Century, Niels Bohr was worrying about the electron orbit of the hydrogen atom, while Albert Einstein was interested in how things appear to moving observers. Bohr’s concern led to the present form of quantum mechanics where the hydrogen atom is a standing wave localized within a finite region. Einstein formulated his special theory of relativity based on the Lorentzian geometry of space and time applicable to Maxwell’s theory of electromagnetism.

It is known that Bohr and Einstein met occasionally to discuss physics. However, there are no written records to indicate that they ever discussed how moving hydrogen atoms appear to a stationary observer. If they did not discuss this problem, it is understandable because there are no observable hydrogen atoms moving with relativistic speeds. Yet, this Bohr-Einstein issue defines an important problem in quantum mechanics. The bound state in quantum mechanics with discrete energy levels is well understood. However, how would those energy levels appear to moving observers? What will happen to the size of the bound state? Indeed, the Bohr-Einstein issue of the hydrogen atom leads to the problem of moving quantum bound states in Einstein’s Lorentz-covariant world.

There are a number of key questions on the moving bound state. In the Lorentz-covariant world, the time variable is linearly mixed with the longitudinal coordinate. There is also the time-energy uncertainty relation. How is this relation mixed with Heisenberg’s uncertainty for momentum and space? Paul A. M. Dirac raised these questions in 1927, and attempted to find a solution using harmonic oscillator wave functions in 1945. In addition, in 1949, he introduced the light-cone coordinate system for squeeze transformations in the two-dimensional space of the time and longitudinal coordinate.

The Bohr radius is a spatial separation between the proton and electron in the hydrogen atom. If this atom is boosted, this spatial separation picks up its time-like component. However, this time-like separation is not included in the present form of quantum mechanics. On the other hand, it is still possible to regard this time separation as an un-observable variable and treat it statistically, using the density matrix [1] - [11]. Then, there comes the question of entropy and temperature from this statistical treatment.

While there are no observable hydrogen atoms, these days, high-energy accelerator produce protons moving with speeds close to that of light. Furthermore, thanks to Gell-Mann’s quark model [12], the proton was found to be a quantum bound state just like the hydrogen atom. Its constituents are the quarks. Since the proton and hydrogen atom share the same bound-state quantum mechanics, it is possible to study moving hydrogen atoms by looking at moving protons.

In 1969, Feynman observed that the ultra-fast proton appears like a collection of an infinite-number of free particles with a wide-spread momentum distribution [13] [14] [15]. Feynman called them partons. The question then is whether Gell-Mann’s quarks and Feynman’s partons are two different ways of observing the same entity. Indeed, the problem of moving hydrogen atom becomes the quark-parton puzzle. The Bohr-Einstein issue of moving hydrogen atom can be addressed in terms of the quark-parton puzzle of Gell-Mann and Feynman, as illustrated in Figure 2.

In this paper, we review first efforts made in the past to resolve this quark-parton issue [16] [17], using the Lorentz-covariant oscillator wave function. We then use the same wave function to study the problem arising from the un-observable time-separation variable.

Paul A. M. Dirac made his lifelong efforts to construct a localized wave function in Einstein’s Lorentz-covariant world. For this purpose, Dirac published

four important papers [18] [19] [20] [21]. By integrating the first three of these four papers, it is possible to construct the harmonic oscillator wave functions which can be Lorentz-transformed [22] [23] [24].

In order to carry out this integration, we need the mathematical instrument constructed by Eugene P. Wigner in his 1939 paper on the inhomogeneous Lorentz group [25]. In his paper, Wigner pointed out a particle in the Lorentz-covariant world has its four-momentum. In addition, this particle has internal space-time symmetries.

Thus in Section 2, we review the aspects of Wigner’s paper applicable to the internal space-time symmetries in Einstein’s Lorentz-covariant world.

In Section 3, we list the first three papers Dirac published from 1927 to 1945 [18] [19] [20], and integrate them. The result is a harmonic oscillator wave function which can be Lorentz-boosted. The time-separation variable plays a prominent role in this Lorentz-covariant wave function.

In Section 4, we examine Feynman’s attempts to construct Lorentz-covariant oscillator wave function starting from a Lorentz-invariant wave equation [26]. Let us consider a hadron (bound state of the quarks) consisting of two quarks. This hadron has two space-time coordinate systems. One is for the hadron moving freely, and the other is for the motion of quarks inside the hadron. For the hadronic coordinate, the Klein-Gordon equation and it solutions are applicable. For the internal coordinate, we can use the harmonic oscillator wave functions constructed from the integration of Dirac’s three papers discussed in Section 3.

In Section 5, this covariant harmonic oscillator is applied to the physics of hadrons. This wave function allows us to Lorentz-boost the hadron at rest to its speed very close to that of light. The hadron at rest is like a quantum bound state like the hydrogen atom according to Gell-Mann’s quark model [12]. However, the same hadron appears like a collection of free massless particles called partons. This aspect is called Feynman’s parton picture of the hadron [13] [14] [15]. The question then is whether the quarks and partons are two different ways of looking at the same entity. We resolve this issue using the Lorentz-covariant oscillator wave functions constructed in Sections 3 and 4. This Lorentz-covariant wave function depends on the time-separation variable which becomes more prominent as the hadron gains its speed.

In Section 6, it is noted that there is a time-separation variable between the quarks. This variable becomes more prominent when the hadron becomes faster. This time-separation is not a measurable dynamical variable in the present form of quantum mechanics. However, the density matrix tells us how to deal with this unobservable variable. It allows us to translate our inability to measure this variable into entropy and temperature. This problem was discussed in the literature [27] [28]. It is shown there that the hadron, when Lorentz boosted, experiences the rise in entropy and also the rise in temperature. It is possible to calculate them as functions of the hadronic speed using the density matrix.

In 1939, Eugene Paul Wigner published his paper entitled On unitary representations of the inhomogeneous Lorentz group [25]. In this paper, Wigner spells out the internal space-time symmetries of particles in the Lorentz-covariant world [22]. Let us consider a particle in this world. It has its four-momentum. If this particle is at rest, it has its rotational degree of freedom. If its spin is 1/2, the symmetry group is SU(2) like (locally isomorphic to) O(3) (three-dimensional rotation group). If its spin is one, its symmetry group is O(3). This aspect is well known.

Massless particles cannot be brought to their rest frames. According to Wigner [25], the little group for the massless particle is like E(2) or the two-dimensional Euclidean group, with one rotational degree of freedom plus two translational degrees of freedom. The rotational degree of freedom can easily be identified with the helicity of the massless particle. However, the two translational degrees have a stormy history until 1987, when Kim and Wigner noted that the E(2) group is like the cylindrical group where both the translations perform up-down translations on the cylindrical surface. This allows us to identify this up-down translation as with the gauge transformation [29].

In the Lorentz-covariant world, Einstein’s momentum-energy relation is applicable to both massive and massless particles, as shown in Table 1. When the massive particle is Lorentz-boosted, its energy-momentum becomes that of the massless particle when its speed becomes very close to that of light.

We are thus led to the question of whether there exists one little group which can be the O(3)-like group when the particle is at rest and the E(2)-like cylindrical group when the particle moves with the speed very close to that of light.

Let us outline the procedure in which the O(3) symmetry becomes that of the cylindrical group. The three-dimensional rotation group is generated by

L x = − i ( y ∂ ∂ z − z ∂ ∂ y ) ,   L y = − i ( z ∂ ∂ x − x ∂ ∂ z ) ,   L z = − i ( x ∂ ∂ y − y ∂ ∂ x ) , (1)

satisfying the commutation relations

[ L x , L y ] = i L z ,   [ L y , L z ] = i L x ,   [ L z , L x ] = i L y . (2)

Let us go to Figure 3. The circle in this figure illustrates the O(3)-like little group for the massive particle at rest.

If it is boosted along the z-direction, the z coordinate picks up the time-like component, and the geometry is four-dimensional. While this geometry is described in detail in the 1987 paper of Kim and Wigner [29], we give here a simplified version.

When the particle is boosted along the z direction, the z component of this circle becomes expanded. If the speed becomes close to that of light, the sphere becomes a cylinder, as indicated in Figure 3.

On the surface of this cylinder, there are no variations of the x and y components, and thus

L x → P y = − i ( y ∂ ∂ z ) ,   L y → − P x = i ( x ∂ ∂ z ) , (3)

and L z remains unchanged. Both P x and P y generate translations along the z direction. These new operators satisfy the commutation relations

[ P x , P y ] = 0 ,   [ L z , P x ] = i P y ,   [ L z , L y ] = − i P x . (4)

It is appropriate to call the group generated by these three operators the cylindrical group.

This set of commutation relation identical with that for E(2) or the two-dimensional Euclidean group with L z as the generator of rotations and P x and P y as the generators of translations along the x and y directions respectively. The translation generators take the form

P x = − i ∂ ∂ x ,   P y = − i ∂ ∂ y . (5)

Thus, the cylindrical group is like (locally isomorphic to) the E(2) group. In Wigner’s original paper, the little group for massless particles is like the E(2) group. The E(2) group can now replaced with the cylindrical group [29] [30].

Let us go back to Figure 3. We would expect the sphere, when Lorentz-boosted, becomes contracted like a pancake according to Einstein’s space contraction. However, this figure shows the opposite effect. The Lorentzian geometry is four-dimensional and the time-like direction should also be included. In this geometry, the Lorentz boost leads to both pancake-like contraction and football-like elongation. The contraction produces the E(2) geometry and the elongation produces the cylindrical geometry as shown in Figure 3. The cylindrical geometry leads to the correct interpretation of the internal space-time symmetry of massless particles.

Paul A. M. Dirac made his lifelong effort to formulate quantum mechanics consistent with Einstein’s special relativity. The Dirac equation of electrons and positrons is a case in point. This equation is well known.

In addition, he made efforts to formulate a mathematical device to deal localized quantum distributions, such as the hydrogen atom, in Einstein’s Lorentz-covariant world. For this purpose, he published the following four papers.

1) In 1927, Dirac pointed out that the time-energy uncertainty should be considered if the system is to be Lorentz-covariant [18].

2) In 1945, Dirac said the Gaussian form could serve as a representation of the Lorentz group [19].

3) In 1949, when Dirac introduced both his instant form of quantum mechanics and his light-cone coordinate system [20], he clearly stated that finding a representation of the inhomogeneous Lorentz group was the task of Lorentz-covariant quantum mechanics.

4) In 1963, Dirac used the symmetry of two coupled oscillators to construct the O ( 3,2 ) de Sitter group, namely the Lorentz group applicable to the three-dimensional ( x , y , z ) space plus two time variables [21]. This paper serves as a prelude to the synthesis of quantum mechanics and special relativity [24] [31] [32].

Dirac’s papers are poetic, mathematically transparent, and easy to understand. This does not necessarily mean that there is nothing to add to his papers. His papers do not have figures. Thus it is profitable to translate his poems into figures and illustrations. Each of the above four papers is independent. Thus it is profitable to connect his 1945 paper to his early paper of 1927, and his 1949 paper to his earlier papers of 1927 and 1945. Furthermore, Eugene Wigner was his brother-in-law. Wigner published an important paper is 1939 providing the mathematical tool for Dirac’s problems, but he never used Wigner’s mathematics in any meaningful ways.

Let us consider what we can add to his papers in order to construct quantum mechanics valid in Einstein’s Lorentz-covariant world. We are particularly interested in how to Lorentz-boost localized wave functions.

In Section 3, we studied the quantum bound state in the Lorentz-covariant world using harmonic oscillator wave functions. This wave function is a representation of Wigner’s little group, which is a subgroup of Dirac’s inhomogeneous Lorentz group.

For free particles in the covariant world, we use the Klein-Gordon equation. If the particle has a space-time extension, it is possible to use harmonic oscillators as we did in Section 3. The question then is whether it is possible to write an equation for both. Indeed, this problem was recognized by Feynman, Kislinger, and Revndal in 1971 [26].

Let us start with two quarks. We are quite familiar with the Klein-Gordon equation for a free particle in the Lorentz-covariant world. We shall use the four-vector notations

x μ = ( x ,   y ,   z ,   t ) ,     and     x μ 2 = x 2 + y 2 + z 2 − t 2 . (16)

Then the Klein-Gordon equation becomes

( − [ ∂ ∂ x μ ] 2 + m 2 ) ϕ ( x ) = 0. (17)

The solution of this equation takes the familiar form

exp [ ± i ( p 1 x + p 2 y + p 3 z ± E t ) ] , (18)

with E = p 1 2 + p 2 2 + p 3 2 + m 2 .

In 1971, Feynman et al. considered two particles a and b bound together by a harmonic oscillator potential, and wrote down the equation [26]

{ − [ ∂ ∂ x a μ ] 2 − [ ∂ ∂ x b μ ] 2 + ( x a μ − x b μ ) 2 + m a 2 + m b 2 } ϕ ( x a μ , x b μ ) = 0. (19)

The bound state of these two particles is one hadron. The constituent particles are called quarks. We can then define the four-coordinate vector of the hadron as

X = 1 2 ( x a + x b ) , (20)

and the space-time separation four-vector between the quarks as

x = 1 2 2 ( x a − x b ) . (21)

Then Equation (19) becomes

{ − [ ∂ ∂ X μ ] 2 + m 0 2 + ( − [ ∂ ∂ x μ ] 2 + x μ 2 ) } ϕ ( X , x ) = 0. (22)

This differential equation can then be separated into

( − [ ∂ ∂ X μ ] 2 + m 0 2 ) ϕ ( X , x ) = − ( − [ ∂ ∂ x μ ] 2 + x μ 2 ) ϕ ( X , x ) , (23)

with

ϕ ( X , x ) = f ( X ) ψ ( x ) , (24)

where f ( X ) and ψ ( x ) satisfy their own equations:

( − [ ∂ ∂ X μ ] 2 + m a 2 + m b 2 + λ ) f ( X ) = 0 (25)

and

1 2 ( − [ ∂ ∂ x μ ] 2 + x μ 2 ) ψ ( x ) = λ ψ ( x ) . (26)

Here, the wave function takes the form

ϕ ( X , x ) = ψ ( x ) exp [ ± i ( P x X + P y Y + P z Z ± E T ) ] , (27)

where P x , P y , P z are for the hadronic momentum, and

E 2 = P x 2 + P y 2 + P z 2 + M 2 ,     with     M 2 = m a 2 + m b 2 + λ . (28)

Here the hadronic mass M is determined by the parameter λ , which is the eigenvalue of the differential equation for ψ ( x ) given in Equation (26).

Considering Feynman diagrams based on the S-matrix formalism, quantum field theory has been quite successful. It is, however, only useful for physical processes where, after interaction, one set of free particles becomes another set of free particles. The questions of localized probability distributions and their Lorentz covariance is not addressed by the present form of quantum field theory. In order to tackle this problem and address these questions, Feynman et al. suggested harmonic oscillators [26]. In Figure 6, we illustrate this idea.

However, for their wave function ψ ( x ) , Feynman et al. uses a Lorentz-invariant exponential form

exp ( − 1 2 [ x 2 + y 2 + z 2 − t 2 ] ) . (29)

This wave function increases as t becomes large. This is not an acceptable wave function. They overlooked the normalizable exponential form given by Dirac in Equation (6). They even overlooked the same normalizable in the paper of Fujimura et al. [34] which was quoted in their own paper.

Thus, we are fully justified in replacing the meaningless Gaussian form of Equation (29) with the Gaussian form developed in Section 3.

Early successes in the quark model include the calculation of the ratio of the neutron and proton magnetic moments [35], and the hadronic mass spectra [26] [36]. These are based on hadrons at rest. We are interested in this paper how the hadrons in the quark model appear to observers in different Lorentz frames.

These days, modern particle accelerators routinely produce protons moving with speeds very close to that of light. Therefore, the question is whether the covariant wave function developed in Section 4 can explain the observed phenomena associated with those protons moving with relativistic speed.

The idea that the proton or neutron has a space-time extension had been developed long before Gell-Mann’s proposal for the quark model [12]. Yukawa [37] developed this idea as early as 1953, and his idea was followed up by Markov [38].

Hofstadter [39] [40], by using electron-proton scattering to measure the charge distribution inside the proton, made the first experimental discovery of the non-zero size of the proton. If the proton were a point particle, the scattering amplitude would just be a Rutherford formula. However, Hofstadter found a tangible departure from this formula which can only be explained by a spread-out charge distribution inside the proton.

Indeed, the first success of the Lorentz-covariant oscillator was demonstrated in the calculation of the Hofstadter effect. Using this wave function, Markov made his calculation in 1956 even before Gell-Mann formulated his quark model in 1964 [38]. After the quark model, many authors made their calculations of the Hofstadter effect using the same wave function [34] [41] [42] [43] [44] [45]. These papers amply demonstrate the elliptic deformation of the Gaussian distribution shown in Figure 4 and Figure 5.

Next, we are facing a more fundamental question. Let us go back to Figure 2. The quark model and its Lorentz-covariant wave function allow us to address the issue of the quark-parton puzzle, and thus the Bohr-Einstein issue of how the hydrogen atom appears to moving observers.

The entropy is a measure of our ignorance and is computed from the density matrix [1] - [11]. The density matrix is needed when the experimental procedure does not analyze all relevant variables to the maximum extent consistent with quantum mechanics. The purpose of the present section to discuss a concrete example of the entropy arising from our ignorance in relativistic quantum mechanics formulated in Sections 3 and 4.

Let us consider a bound state of two particles, or the hadron consisting of two quarks bound together by a harmonic oscillator potential. Then there is a Bohr-like radius measuring the space-like separation between the quarks. There is also the time-like separation in the Lorentz-covariant regime. If the hadron is at rest, the time dependence is purely Gaussian with no excitations. Thus, this un-observable variable can be integrated out without affecting the space-like separation.

If the hadron moves along the z direction, this time-separation variable becomes more prominent, but there are no ways to measure this variable in the present form of quantum mechanics. We thus have to regard this variable as un-measurable variable, and treat it statistically.

As in the case of Section 4, let us consider a hadron consisting of two quarks. If the space-time position of two quarks are specified by x a and x b respectively, the system can be described by the variables

X = x a + x b 2 ,     and     x = x a − x b 2 2 . (38)

The four-vector X specifies where the hadron is located in space and time, while the variable x measures the space-time separation between the quarks. In the convention of Feynman et al [26], the internal motion of the quarks bound by a harmonic oscillator potential can be described by the Lorentz-invariant equation

1 2 { [ − ∂ 2 ∂ z 2 + z 2 ] − [ − ∂ 2 ∂ t 2 + t 2 ] } ψ ( z , t ) = λ ψ ( z , t ) . (39)

For simplicity, we do not consider the transverse coordinates x and y.

It is possible to construct a representation of Dirac’s inhomogeneous Lorentz group [22] from the solutions of the differential equation of Equation (39). If the hadron is at rest, the solution should take the form of Equation (8). Let us rewrite this solution as the wave function

ψ 0 n ( z , t ) = ( 1 π n ! 2 n ) 1 / 2 H n ( z ) exp { − [ z 2 + t 2 2 ] } . (40)

The subscript 0 means that the wave function is for the hadron at rest. The above expression is not Lorentz-invariant, and its localization undergoes a Lorentz squeeze as the hadron moves along the z direction as shown in Figure 4.

For this Lorentz-covariant system, it is convenient to use the light-cone variables

u = z + t 2 ,   v = z − t 2 , (41)

introduced in Equation (12). The lorentz-boost along the z axis leads to

u → u ′ = e η u ,   v → v ′ = e − η v , (42)

where η is the boost parameter and is tanh η = v / c . In terms of these light-cone variables, the wave function of Equation (40) can be written as

ψ 0 n ( x , t ) = ( 1 π n ! 2 2 ) 1 / 2 H n ( u + v 2 ) exp ( − [ u 2 + v 2 2 ] ) . (43)

If the system is boosted, the wave function becomes

ψ η n ( x , t ) = ( 1 π n ! 2 2 ) 1 / 2 H n ( e − η u + e η v 2 ) exp ( − [ e − 2 η u 2 + e 2 η v 2 2 ] ) . (44)

This wave function can be expanded as [24]

ψ η n ( z , t ) = ( 1 cosh η ) ( n + 1 ) ∑ k [ ( n + k ) ! n ! k ! ] 1 / 2 ( tanh η ) k χ n + k ( z ) χ k ( t ) , (45)

where χ n ( z ) is the n-th excited-state oscillator wave function.

Here comes the fundamental problem. If the hadron is at rest, this wave function is separable in z and t. If the t variable is integrated out, the rest is the present form of non-relativistic quantum mechanics.

However, if the hadron moves and gains speed, the t dependence becomes non-separable, and we have to resort to density matrix. From the wave function of Equation (6), we can construct the pure-state density matrix

ρ η n ( z , t ; z ′ , t ′ ) = ψ η n ( z , t ) [ ψ η n ( z ′ , t ′ ) ] * , (46)

where [ ψ η n ( z ′ , t ′ ) ] * = ψ η n ( z ′ , t ′ ) . This pure-state density matrix satisfies the condition ρ 2 = ρ :

ρ η ( z , t ; x ′ , t ′ ) = ∫ ρ η ( z , t ; x ″ , t ″ ) ρ η ( z ″ , t ″ ; z ′ , t ′ ) d z ″ d t ″ . (47)

However, there are at present no measurement theories which accommodate the time-separation variable t. Thus, we can take the trace of the ρ matrix with respect to the t variable. Then the resulting density matrix is

ρ η n ( z , z ′ ) = ∫ ψ η n ( z , t ) ψ η n ( z ′ , t ) d t (48)

= ( 1 cosh η ) 2 ( n + 1 ) ∑ k ( n + k ) ! n ! k ! ( tanh η ) 2 k χ n + k ( z ) χ n + k ( z ′ ) . (49)

The trace of this density matrix is one, but the trace of ρ 2 is less than one, as

T r ( ρ 2 ) = ∫ ρ ( z , z ′ ) ρ ( z ′ , z ) d z ′ d z (50)

= ( 1 cosh η ) 4 ( n + 1 ) ∑ k [ ( n + k ) ! n ! k ! ] 2 k ( tanh η ) 4 k . (51)

which is less than one. This is due to the fact that we do not know how to deal with the time-like separation in the present formulation of quantum mechanics. Our knowledge is less than complete.

The standard way to measure this ignorance is to calculate the entropy defined as [1] [2] [3] [27]

S = − T r [ ρ ln ( ρ ) ] . (52)

This formula is known as the Shannon entropy in the current literature on quantum computation and quantum information [11].

If we pretend to know the distribution along the time-like direction and use the pure-state density matrix given in Equation (46), then the entropy is zero. However, if we do not know how to deal with the distribution along the time separation t, then we should use the density matrix of Equation (48) to calculate the entropy, and the result is [27]

S = 2 ( n + 1 ) [ ( cosh η ) 2 ln ( cosh η ) − ( sinh η ) 2 ln ( sinh η ) ] (53)

  − ( 1 cosh η ) 2 ( n + 1 ) ∑ k ( n + k ) ! n ! k ! [ ln [ ( n + k ) ! n ! k ! ] ] ( tanh η ) 2 k . (54)

Let us go back to the wave function given in Equation (6). As is illustrated in Figure 4, its localization property is dictated by the Gaussian factor which corresponds to the ground-state wave function. For this reason, we expect that much of the behavior of the density matrix or the entropy for the n-th excited state will be the same as that for the ground state with n = 0. For this state, the density matrix and the entropy are

ρ ( z , z ′ ) = ( 1 π cosh ( 2 η ) ) 1 / 2 exp { − 1 4 [ ( z + z ′ ) 2 cosh ( 2 η ) − ( z − z ′ ) 2 cosh ( 2 η ) ] } , (55)

and

S = ( cosh 2 η ) ln ( cosh 2 η ) − ( sinh 2 η ) ln ( sinh 2 η ) , (56)

respectively. The quark distribution ρ ( z , z ) becomes

ρ ( z , z ) = [ 1 π cosh ( 2 η ) ] 1 / 2 exp ( − z 2 cosh ( 2 η ) ) . (57)

The width of the distribution becomes cosh ( 2 η ) , and becomes wide-spread as the hadronic speed increases. Likewise, the momentum distribution becomes wide-spread, as in the case of Feynman’s parton picture described in Subsection 5.1. This simultaneous increase in the momentum and position distribution widths is called the parton phenomenon in high-energy physics [13] [14] [15]. The position-momentum uncertainty becomes cosh 2 η . This increase in uncertainty is due to our ignorance about the physical but unmeasurable time-separation variable. This does not violate the fundamental law of the uncertainty as described in Subsection 5.2.

The use of an unmeasurable variable as a shadow coordinate is not new in physics [7] [8]. Feynman’s book on statistical mechanics contains the following paragraph [5].

When we solve a quantum-mechanical problem, what we really do is divide the universe into two parts—the system in which we are interested and the rest of the universe. We then usually act as if the system in which we are interested comprised the entire universe. To motivate the use of density matrices, let us see what happens when we include the part of the universe outside the system.

In the present paper, we have identified Feynman’s rest of the universe as the time-separation coordinate in a relativistic two-body problem. Our ignorance about this coordinate leads to a density matrix for a non-pure state, and consequently to an increase of entropy. Figure 9 shows the entropy as a function of the hadronic speed or tan η . The entropy is zero when the hadron is at rest. It increases rapidly as the hadronic speed approaches the speed of light.

Finally, let us examine how the ignorance will lead to the concept of temperature [28]. For the Lorentz-boosted ground state, the density matrix of Equation (48) becomes

ρ η ( z , z ′ ) = ( 1 cosh η ) ∑ k ( tanh η ) 2 k χ k ( z ) χ k ( z ′ ) . (58)

We can now compare this expression for the oscillator system in the thermally excited state. In terms of the temperature T, the density matrix takes the form [46] [47].

ρ T ( z , z ′ ) = ( 1 − e − 1 / T ) ∑ n     e − n / T χ n ( z ) χ n ( z ′ ) , (59)

where T means k T / ℏ ω , with k and ω as Boltzmann’s constant and the frequency of oscillation respectively. If we compare this expression with Equation (58). Then

e − 1 / T = ( tanh η ) 2 . (60)

This leads to

T = − 1 2 ln ( tanh η ) . (61)

The temperature rises rapidly as the hadronic speed approaches the speed of light, as indicated in Figure 8. As we noted in Subsection 5.1, the hadron becomes a plasma state as in the case of Feynman’s parton picture.

The primary purpose of this paper was to study the entropy and temperature of bound states in the Lorentz-covariant world. The result of study is given in Section 6.

This problem arises because the time separation between the constituent particles is not a measurable quantity in the present form of quantum mechanics. On the other hand, it is possible to treat this unmeasurable variable statistically. The time separation is negligible when the bound state is at rest, but it becomes as significant as the space separation (like the Bohr radius) when the bound state moves.

In order to study this effect, we need at least one Lorentz-covariant model for bound states. Dirac and Feynman made their efforts to construct such a wave function. Much of the present paper is devoted to the integration of their efforts to construct wave functions for moving bound states in Einstein’s Lorentz-covariant world.

It was Paul A. M. Dirac who made efforts to construct wave functions for moving bound states. Dirac’s papers and books are like beautiful poems, but they do not contain figures. It was a challenge to convert his poems [18] [19] [20] into the circle and the rectangle given in Figure 4. Then it is easy to integrate those two figures. A more detailed explanation is given in a recent book entitled Physics of the Lorentz Group, 2nd Edition by Başkal, Kim, and Noz [24].

We used in this paper the Lorentz-covariant wave function which provides the resolution to the question whether the quarks and partons are two different ways of looking at the same entity in the Lorentz-covariant world, as illustrated in Figure 8. This also provides the answer to the Bohr-Einstein issue of moving hydrogen atoms.

This review paper is largely based on the papers and books I, the author of this paper, published with Marilyn Noz since 1973. I thank her for this prolonged collaboration. I thank also my younger colleagues who co-authored with me many papers I published since 1978. Their names are D. Han, P. Hussar, S. H. Oh, and D. Son. My later co-authors include S. Başkal, and E. Georgieva.

In July of 1962, I became an assistant professor of physics at the University of Maryland. Paul A. M. Dirac visited the University for one week in October of the same year at the invitation of John S. Tall who was the chairman of the physics department. Toll assigned me as the personal assistant to Dirac. This gave me an excellent opportunity to learn physics directly from Dirac.

Dirac told me that physicists in general do not understand the difference between the Lorentz invariance and the Lorentz covariance. This difference was illustrated in terms of the hyperbola and the squeezed circle Figure 4. This allowed me to integrate Dirac’s attempts to construct the wave function in the Lorentz-covariant world.

Another fundamental issue is the unification of the internal space-time symmetries. In his paper of 1939 [25], he stated that those symmetries are like O(3) and and E(2) respectively. The question then is whether the E(2) symmetry can be obtained continuously as the mass of particle becomes zero. This question has a stormy history [48]. Because the transition from O(3) to E(2) is not an analytic continuation. I am grateful to Professor Eugene Paul Wigner for spending time with me to settle this issue as described in Section 2 of the present paper.

The author declares no conflicts of interest regarding the publication of this paper.

Kim, Y.S. (2022) Quantum Mechanics of Moving Bound States. Journal of Modern Physics, 13, 138-165. https://doi.org/10.4236/jmp.2022.132012

In Section 4, we noted that the bound state has its internal space-time coordinates, and we considered a bound state equation in Equation (26). Let us write this oscillator equation:

1 2 ( − [ ∂ ∂ x μ ] 2 + x μ 2 ) ψ ( x ) = λ ψ ( x ) . (62)

This differential equation is separable in all space-time variables. Thus we can concentrate on the longitudinal and time coordinates. The equation then becomes

1 2 [ ( − [ ∂ ∂ z ] 2 + z 2 ) − ( − [ ∂ ∂ t ] 2 + t 2 ) ] ψ ( z , t ) = λ ψ ( z , t ) . (63)

Since the time excitations are not allowed, the solution of this equation takes the form

ψ n ( z , t ) = ( 1 π ) 1 / 4 exp ( − t 2 2 ) ξ n ( z ) , (64)

where ξ n ( z ) is the oscillator wave function for the n-th excited state. The differential equation of Equation (63) is invariant under the Lorentz boost:

z → z + β t 1 − β 2 ,   t → t + β z 1 − β 2 , (65)

and the boosted wave function becomes

ψ β n ( z , t ) = ( 1 π ) 1 / 4 exp ( − t ′ 2 2 ) ϕ n ( z ′ ) , (66)

with

z ′ = z − β t 1 − β 2 ,   t ′ = t − β z 1 − β 2 . (67)

The wave function of Equation (66) becomes ψ n ( z , t ) given in Equation (64) when β = 0 . There are no excitations along the t direction because of the c-number time-energy uncertainty relation [18].

It is then of interest to evaluate the integral [24] [49], and the result is

∫ ψ n ( z , t ) ψ β n ′ ( z , t ) d x d t = ( 1 − β 2 ) ( n + 1 ) δ n n ′ . (68)

The orthogonality relation and the contraction property contained in this formula are illustrated in FigureA1. The stationary ground state wave function is orthogonal to all excited states. This ground state is contracted by 1 − β 2 . This is consistent with our understanding of Einstein’s Lorentz contraction of a rod.

Then why is [ 1 − β 2 ] ( n + 1 ) for the n-th excited states? It is because the (wavefunction)² has (n + 1) humps. Then why are they multiplicative?

In order to answer this question, let us use the bra-and-ket notation for the harmonic oscillators, where the ground state is | 0 〉 and | n 〉 is for the n-th excited state. We use a and a † for step-down and step-up operators respectively. As is well known, the n-th excited state becomes

| n 〉 = 1 n ! ( a † ) n | 0 〉 . (69)

Thus, each additional hump is produced through the multiplication process.