Journal of Basic and Applied Research International1(1): 13-23, 2014

THE RELATIVISTIC DYNAMICS AS A QUANTUM EFFECT

ELIADE STEFANESCU∗

Center of Advanced Studies in Physics, Institute of Mathematics Simion Stoilow of the RomanianAcademy, 13 Calea 13 Septembrie, 050711 Bucharest S5, Romania [ES].

[*For Correspondence: E-mail: eliadestefanescu@yahoo.fr]

Abstract

In this paper we develop a Unitary Quantum Relativistic Theory. We show that the wavy natureof a quantum particle involves the relativistic dynamics of the wave-packet of this particle, anda field of interaction described by Lorentz’s force and three of the four Maxwell equations: theelectromagnetic induction law, and the flow laws of the electric and magnetic fields. Theseequations arise from the group velocity, which is of the form of the Lagrange equation, andfrom a relativistic principle for wave functions, which asserts that in any system of referencea wave-function has a bounded spectrum with a velocity limit c. When a magnetic circuit lawis considered for a field interacting with a quantum particle, this is an electromagnetic field,propagating with the velocity c.

Keywords: Wave-function, wave-packet, group velocity, Hamiltonian, Lagrangian, Lorentz force,Hamilton equations, Lagrange equation, Lorentz transformation, Maxwell equations.

1 IntroductionRelativity is a theory of space and time, essential for understanding the classical dynamics (Einstein,2008; Mermin, 2009; Feynman, 1964), new domains of research (Smolin, 2006; Barrow, 2007;Penrose, 2004; Hawking, 2010) and, generally, our physical world (Bohm, 2006; Howking, 1993;Bernstein, 1988; Schiller, 1990). However, this space-time theory is not taken at the basis of quantummechanics, which is essential for many applications standing at the basis of our civilization. The useof the quantum theory encounters the difficulty that it involves ingredients of the classical theoriesas the Newtonian mechanics and the Maxwellian electromagnetism. More than that, when a ’non-relativistic’ quantum theory is considered for a matter system coupled to an electromagnetic field, thistheory becomes in fact half-relativistic, since the electromagnetic field is generated by the theory ofrelativity (Einstein, 1905; Landau & Lifchitz, 1966; Fock, 1964), which means a certain inconsistencyin the description of such a coupling. For instance, in a previous paper (Stefanescu, 2014), westudied a quantum device based on the matter-field interaction. A basic element of this approach isthe canonic momentum operator of an electron in electromagnetic field,

P = p+ eA(r, t) , (1.1)

where p is the electron momentum operator, e is the electron charge, and A(r, t) is the vector potentialoperator of the electromagnetic field. The problem is that the canonic momentum (1.1) is a sum of aclassical term with a relativistic one. Another problem is that this operator is obtained from a classicalLagrangian constructed to satisfy the classical Lorentz’s equation,

d

dtp = eE(r, t) + e ˙r × B(r, t) , (1.2)

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and only after that the classical quantities are replaced by operators. The usual procedure of ’quantiza-tion’, consisting in the consideration of the classical or relativistic equations and only after thatimposing quantum conditions as the ’uncertainty’ relations, could be taken only as a technique, butnot as representing the essence of the physical phenomena. To avoid the derivation of equation(1.1) from the classical equations, which in quantum mechanics are obtained for mean-values, in(Stefanescu, 2014) we considered a wave-function as a wave-packet, with a conjugated wave-vector1~ p of the position vector r:

ψ(r, t) =1

(2π~)3/2

∫ϕ(p, t)e

i~ {pr−[T (p)−U(r)]t}d3p , (1.3)

where T (p) and U(r) are scalar functions of the two vectors. Taking into account also the inversetransformation of this Fourier integral,

ϕ(p, t) =1

(2π~)3/2

∫ψ(r, t)e−

i~ {pr−[T (p)−U(r)]t}d3r , (1.4)

we obtained the group velocities of the two wave-packets of the form of the Hamiltonian equations

d

dtr =

∂

∂pT (p) =

∂

∂pH(r, p) (1.5a)

d

dtp = − ∂

∂rU(r) = − ∂

∂rH(r, p) , (1.5b)

whereH(r, p) = T (p) + U(r) (1.6)

is the Hamiltonian of a non-relativistic particle.In this paper, we develop a similar theory for a relativistic particle. In section II, we describe a

quantum particle as a wave-packet with coordinates which satisfy the Lorentz transformation, whilethe energy is obtained in agreement with the famous formula E = Mc2. In section III, we considerthe interaction with a field. From the group velocity, we obtain the Lorentz dynamic equation, andfield equations. In section IV, we obtain relativistic field transformations. In section V, we obtainfield invariants and define the electromagnetic field by the Ampre-Maxwell equation, while the otherfield equations are general characteristics of the particle-field interaction. In section VI, we giveconclusions.

2 Relativity Principle for a Wave-FunctionThe relativistic dynamics of particle has originally been obtained by Einstein in the framework of aclassical electromagnetic theory (Einstein, 1905). A classical treatment of this problem is based onon the least action principle, where classical paths are considered (Landau & Lifchitz, 1966). In thispaper, we avoid such classical paths, which are not conceivable for a quantum particle. With theHamilton equations (1.5), we obtain the Hamiltonian (1.6) as a conservative function:

d

dtH(r, p) =

∂H

∂r

dr

dt+∂H

∂p

dp

dt= 0 . (2.1)

Taking p as the mechanical momentum, while ~ is Planck’s action, we get the particle energy E asthe value of the Hamiltonian function:

H(r, p) = E . (2.2)

According to (1.6), the total energy E of the particle is the sum of the kinetic energy T (p) with thepotential energy U(r). With a classical momentum, as a product of the particle mass with the velocity˙r,

p =M ˙r , (2.3)

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from the Hamilton equation (1.5a) , we get a momentum equation

p =M∂

∂pT (p) , (2.4)

which, by integration, leads to the classical kinetic energy

T (p) =p2

2M. (2.5)

At the same time, the variation of the Hamiltonian function is of the form

δH =∂H

∂rδr +

∂H

∂pδp = − ˙pδr + ˙rδp , (2.6)

whileδ(p ˙r) = ˙rδp+ pδ ˙r . (2.7)

From these equations, we get the Lagrangian

L(r, ˙r) = p ˙r −H , (2.8)

with the variationδL = ˙pδr + pδ ˙r =

∂L

∂rδr +

∂L

∂ ˙rδ ˙r , (2.9)

which leads to the Lagrange equation:

˙p =d

dt

∂L

∂ ˙r=∂L

∂r. (2.10)

With (1.6), (2.3), and (2.5), for the Lagrangian (2.8), we find the classical expression

L(r, ˙r) = T (p)− U(r) =M ˙r2

2M− U(r) , (2.11)

which leads to a particle wave-function (1.3) of a more compact form

ψ(r, t) =1

(2π~)3/2

∫ϕ( ˙r, t)e

i~ [M ˙rr−L(r, ˙r)t]M3d3 ˙r . (2.12)

We notice that although this wave-function is in agreement with the Hamilton equations (1.5) andthe Lagrange equation (2.10), due to the classical expression (2.11) it is still unphysical, includingcomponents with infinitely high velocities, and infinitely high energies: a kind of ultraviolet explosion.For a free particle, this unphysical effect can be avoided considering a Lagrangian decreasing to zerofor a rather high velocity | ˙r| ∼ c,

L = −M0c2

√1−

˙r2

c2, (2.13)

which, for small velocities, | ˙r| ≪ c, leads to a momentum in agreement with the classical one:

p =∂L

∂ ˙r=

M0˙r√

1− ˙r2

c2

≈M0˙r∣∣∣| ˙r|≪c

. (2.14)

In this case, we obtain the group velocity (figure 1a)

˙r =∂L

M∂ ˙r=

p

M. (2.15)

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a:-

ddt r = ∂

∂pH(r, p)

M ddt r = p = ∂L

∂ ˙r

b:

0 x x

x′x′

τ

τ

τ ′

τ ′

φ

φ

Figure 1: a: Wave-packet with a group velocity as a Hamilton equation, or an equation ofLagrange;

b: Lorentz transformation as a rotation of the space-time system.

From these equations, we obtain the relativistic mass

M =M0√1− ˙r2

c2

, (2.16)

From these equations, we obtain the relativistic mass

M =M0√1− ˙r2

c2

, (2.17)

as a function of the particle mass at low velocities M0 and the velocity ˙r. It is remarkable that thewave-function (2.12) with the Lagrangian (2.13) of a particle is the same in any system of reference,which means that the differential interval

ds = c

√1−

˙r2

c2dt =

√c2dt2 − dx2 − dy2 − dz2 (2.18)

is invariant. According to a usual approach of the relativistic theory, in a referential {x, y, z, τ = ict}, areferential {x′, y′, z′, τ ′ = ict′}, moving in the x-direction with a velocity V , is described by a rotation

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with an angle φ (figure 1b),

tanφ = −xτ

∣∣∣(x,τ)∈τ ′-axis

= − x

ict= i

V

c. (2.19)

Thus, one obtains the transformation relations

x = x′ cosφ− τ ′ sinφ (2.20)

τ = τ ′ cosφ+ x′ sinφ , (2.21)

which lead to the Lorentz transformation:

x =x′ + V t′√1− ˙r2

c2

(2.22)

y = y′ (2.23)

z = z′ (2.24)

t =t′ + V

c2x′√

1− ˙r2

c2

. (2.25)

In this framework, for the characteristics the particle wave-packet (2.12) represented in figure 1a,we obtain basic equations of the relativistic theory. Thus, with (2.14), the relativistic mass (2.17) takesthe form of a function of the momentum,

M =1

c

√p2 +M2

0 c2 . (2.26)

With this expression and equations (2.13)-(2.14), from the general equation (2.8) we obtain theHamiltonian

H = c√p2 +M2

0 c2 , (2.27)

and the relativistic energy

E =Mc2 =M0c

2√1− ˙r2

c2

. (2.28)

It is remarkable that these relativistic equations do not refer to a classical particle, but to a quantumparticle described by the wave-function (2.12).

3 Field Interaction with a Quantum ParticleFor a particle with a wave-function (2.12) and an electric charge e, interacting with some other systemaccording to a more general Lagrangian

L(r, ˙r, t) = −M0c2

√1−

˙r2

c2+ eA(r, t) ˙r − eU(r, t) , (3.1)

the momentum takes the classical form (1.1),

P =∂

∂ ˙rL( ˙r, r, t) =

M0˙r√

1− ˙r2

c2

+ eA(r, t) = p+ eA(r, t) , (3.2)

of a quasi-particle with a mechanical momentum (2.14) of the particle, and a component eA(r) of thefield dressing this particle. We notice that the mass M of this quasi-particle, defined by the relation

P =M ˙r =∂

∂ ˙rL(r, ˙r, t) = p+ eA(r, t) , (3.3)

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includes a field component eA(r, t). With this expression, from the inverse transformation of theFourier integral (2.12)

ϕ(M ˙r, t) =1

(2π~)3/2

∫ψ(r, t)e−

i~ [M ˙rr−L(r, ˙r,t)t]d3r , (3.4)

we get a group velocity as a Lagrange equation

d

dt(M ˙r) =

d

dt

∂

∂ ˙rL(r, ˙r, t) =

∂

∂rL(r, ˙r, t) . (3.5)

With (3.1) and (3.2), we get the Hamiltonian

H = P ˙r − L(r, ˙r, t) = c

√M2

0 c2 +

[P − eA(r, t)

]2+ eU(r, t) , (3.6)

as a conservative function with the energy value

E =M0c

2√1− ˙r2

c2

+ eU(r, t) . (3.7)

At the same time, from the Lagrange equation (3.5) with the momentum (3.3), we get the equation ofmotion

d

dtP =

∂

∂rL(r, ˙r, t) = e

∂

∂r

[A(r, t) ˙r

]− e

∂

∂rU(r, t) . (3.8)

With the expression (3.2) of the canonic momentum, and the relations

∂

∂r

[A(r, t) ˙r

]=

[˙r∂

∂r

]A(r, t) + ˙r ×

[∂

∂r× A(r, t)

](3.9)

andd

dtA(r, t) =

∂

∂tA(r, t) +

(˙r∂

∂r

)A(r, t) , (3.10)

we obtain the Lorentz force

Fe =d

dtp =

d

dtP − e

d

dtA(r, t) = eE(r, t) + e ˙r × B(r, t), (3.11)

depending on the electric field

E(r, t) = − ∂

∂rU(r, t)− ∂

∂tA(r, t) , (3.12)

and the magnetic induction

B(r, t) =∂

∂r× A(r, t) . (3.13)

With these expressions, we obtain the electromagnetic induction law

∂

∂r× E(r, t) = − ∂

∂tB(r, t) , (3.14)

and the magnetic flow law:∂

∂rB(r, t) = 0 . (3.15)

At the same time, with the gouge condition

∂

∂rA(r, t) = 0, (3.16)

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from (3.12) we get the electric field flow equation

∂

∂rE(r, t) =

ρ(r, t)

ε0, (3.17)

where we defined the charge density as a potential source

ρ(r, t) = −ε0∂2

∂r2U(r, t). (3.18)

Defining the charge flow density asj(r, t) = ˙rρ(r, t) , (3.19)

the electric field flow equation (3.17) takes the form

ε0d

dtE(r, t) = j(r, t) + ε0

∂

∂tE(r, t). (3.20)

4 Lorentz Transformation of a FieldSince the wave-function (2.12) with the Lagrangian (3.1) is the same in any system of reference, thefrequency term of this wave-function is an invariant, i.e. with a null variance:

δ[L(r, ˙r, t)dt] = −M0cδds+ eδ[A(r, t)dr]− eδ[U(r, t)dt] = 0 , (4.1)

whereds =

√c2dt2 − dr2 . (4.2)

In this case, the variation of the whole action in an arbitrary interval of time is null:

δS =

∫ {−M0cδds+ eδ[A(r, t)dr]− eδ[U(r, t)dt]

}= 0 . (4.3)

With a four-vector notation (Landau & Lifchitz, 1966),

(xi) = (x, y, z, ict) (4.4)

ds =√

−dx2i (4.5)

(Ai) = (Ax, Ay, Az,i

cU) , (4.6)

the the action of the Lagrangian (3.1) in a differential of time takes a simpler form

L(r, ˙r, t)dt = −M0c2ds+ eAidxi . (4.7)

Calculating the variations

δds = δ√

−dx2i = −dxids

δdxi (4.8)

δ(Aixi) = Aiδdxi + δAidxi , (4.9)

equation (4.3) becomes:

δS =

∫ {M0c

dxids

δdxi +Aiδdxi + δAidxi

}= 0 . (4.10)

Considering the velocity four-vector

ui =dxids

withd

ds=

∂

∂xk

dxkds

= uk∂

∂xk, (4.11)

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and integrating by parts the first two terms, this integral takes a form∫ {−M0c

dui

ds+ e

[∂Ak

∂xi− ∂Ai

∂xk

]uk

}dsδxi = 0 . (4.12)

This equation leads to a system of dynamical equations

M0cdui

ds= eFikuk , (4.13)

depending on the field four-tensor

Fik =∂Ak

∂xi− ∂Ai

∂xk. (4.14)

We notice that this tensor is antisymmetric, with elements depending on the components (3.12) and(3.13) of the electromagnetic field,

F11 = F22 = F33 = F44 = 0 (4.15)

F12 = −F21 = Bz , F13 = −F31 = −By , F14 = −F41 = − i

cEx (4.16)

F23 = −F32 = Bx , F24 = −F42 = − i

cEy (4.17)

F34 = −F43 = − i

cEz , (4.18)

has the matrix form:

(Fik) =

0 Bz −By − i

cEx

−Bz 0 Bx − icEy

By −Bx 0 − icEz

icEx

icEy

icEz 0

. (4.19)

With these expressions, from the dynamical equations (4.13), we reobtain the relativistic Lorentz force(3.11):

d

dt

(M ˙r

)= eE + e ˙r × B (4.20)

and the energy conservation equation of a particle in electromagnetic field

d

dt(Mc2) = eE ˙r . (4.21)

For the action variance (4.12), we consider the Lorentz transformation of the space-time system (x′i)moving with a velocity V in the x-direction of the system (xi),

xi = αijx′j , (4.22)

where, with the notation

γ =1√

1− V 2

c2

, (4.23)

the transformation matrix is of the form

(αij) =

γ 0 0 −iγ V

c

0 1 0 00 0 1 0

iγ Vc

0 0 γ

, (α−1ij ) =

γ 0 0 iγ V

c

0 1 0 00 0 1 0

−iγ Vc

0 0 γ

= (αji) . (4.24)

We obtain ∫(−M0cduiδxi + Fikukδxids) =

∫ (−M0cdu

′iδx

′i + F ′

iku′kδx

′ids

)= 0 . (4.25)

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For the first term, this transformation is an identity,

duiδxi = αijαikdu′jδx

′k = α−1

ji αikdu′jδx

′k = δjkdu

′jδx

′k = du′

jδx′j , (4.26)

while for the second term of this transformation, we obtain the relations

Fikukδxi = Fikαklu′lαijδx

′j = F ′

jlu′lδx

′j , (4.27)

which meanF ′jl = Fikαijαkl , (4.28)

orFik = F ′

jlα−1ji α

−1lk = αijαklF

′jl . (4.29)

From this relation with the matrix (4.24), we find the Lorentz transformation of the field

Ex = E′x (4.30)

Ey = γ(E′y + V B′

z) (4.31)

Ez = γ(E′z − V B′

y) (4.32)

Bx = B′x (4.33)

By = γ(B′y − V

c2E′

z) (4.34)

Bz = γ(B′z +

V

c2E′

y) , (4.35)

which in a more compact representation is of the form:

E = E′ − γV × B′ (4.36)

B = B′ +γ

c2V × E′ . (4.37)

5 Invariants and Electromagnetic FieldIt is remarkable that the field four-tensor (4.19) has two invariants (Landau & Lifchitz, 1966),

FikFki = F ′ikF

′ki (5.1)

eijklFijFkl = eijklF′ijF

′kl , (5.2)

where eijkl is the unity anti-symmetric four-tensor. The first invariant is obtained from a directapplication of the Lorentz transformation:

FikFki = αijαklF′jlαkmαinF

′mn = α−1

ni αijα−1mkαklF

′jlF

′mn = F ′

jlF′lj . (5.3)

The second invariant comes from the Lorentz transformations of the field four-tensor, and of the unityanti-symmetric four tensor:

eijkl = αisαjuαkvαlwesuvw = esuvwα−1si α

−1uj α

−1vk α

−1wl . (5.4)

We get:

eijklFijFkl = esuvwα−1si α

−1uj α

−1vk α

−1wl αimαjnαkpαlqF

′mnF

′pq = emnpqF

′mnF

′pq . (5.5)

With (4.19), the two invariants (5.1) and (5.2) take the explicit forms:

1

c2E2 − B2 =

1

c2E′2 − B′2 (5.6)

EB = E′B′ . (5.7)

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At the same time, from (4.36) and (4.37), we obtain

(V E′)(V B′) = V 2(E′B′) (5.8)1

c2(V E′)2 − (V B′)2 = V 2

(1

c2E′2 − B′2

), (5.9)

which accepts a solution E′ ⊥ B′ ⊥ V . We notice that the two perpendicular fields E and Binteracting with a quantum particle are connected by equation (3.14) of the electromagnetic induction,where the circulation of the electric field on a closed curve is generated by the time variation of themagnetic field flow through a surface bounded by this curve. Considering a symmetric relation for thecirculation of the magnetic field on a closed curve, as a time variation of the electric field flow througha surface bounded by this curve in equation (3.20), we re-obtain the Ampere-Maxwell law:

1

µ0

∂

∂r× B(r, t) = j(r, t) + ε0

∂

∂tE(r, t). (5.10)

In this way, we obtain the Maxwell equations (5.10), (3.14), (3.15), and (3.17) as characteristics ofa field interacting with a quantum particle with a wave-function (2.12). Equations (3.14), (3.15), and(3.17) are general characteristics of the quantum/wavy nature of a particle, while equation (5.10) isa specific characteristics of the electromagnetic field, which propagates with the limit velocity c of aparticle quantum spectrum.

6 ConclusionsIn this paper, we obtained the relativistic dynamics, and the electromagnetic field as characteristicsof a quantum particle. This description is based on a presentation of a quantum particle through itswave function that includes the particle mechanical characteristics and its electromagnetic potentials,while the group velocity is of the form of a Lagrange equation. It is remarkable that a quantum particleis described by a wave-function with a bounded spectrum, with the same velocity limit c in any systemof reference. How strange this could be, it is the nature of our world. It is interesting that threeof the Maxwell equations are general characteristics of the field interaction with a quantum particle,while only the fourth equation, namely the Ampere-Maxwell law, is specific to the electromagneticfield. The dynamic equation of Lorentz’s force is also a general characteristic of a field interactingwith a quantum particle. The characteristic functions of a field are characteristics of the interactionof this field with a quantum particle. A field is characterized by three functions conjugated to thethree coordinates, which are called ’vector potential’, and a potential conjugated to time, which iscalled ’electric potential’. From these potentials, we obtained the magnetic field as a component ofthe electromagnetic field without any source, null divergence. With a gouge condition, we obtain theelectric component of the electromagnetic field with a source given by the Laplacian of the electricpotential, which is called electric charge density. In the framework of this theory, the dynamics of aparticle wave-function is described as a function of the dynamics the field embedding this particle.The classical state functions and the interaction fields are characteristics of this wave-function.

Competing InterestsThe authors declare that no competing interests exist.

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