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Review articleVolume 4 Issue 4 - December 2018
Ann Rev ResearCopyright © All rights are reserved by Eliade Stefanescu
Quantum Mechanics as a Fourier Representationof a Relativistic Quantum Particle
Eliade Stefanescu*Advanced Studies in Physics Centre of the Romanian Academy, Romania
Submission: September 09, 2018; Published: December 17, 2018*Corresponding author: Eliade Stefanescu, Advanced Studies in Physics Centre of the Romanian Academy, Academy of Romanian Scientists, Bucharest, Romania
Ann Rev Resear 4(4): ARR.MS.ID.555641 (2018) 0087
Annals ofReviews and Research
Abstract
The starting point of this research is a representation of a quantum particle according to the Schrodinger equation of the conventional quantum mechanics. In this representation, a quantum particle is described by a wave packet in the coordinate space and the conjugated wave packet in the momentum space. The problem is that while in the coordinate space the group velocity is in agreement with one of the Hamilton equations, the group velocity in the momentum space is in contradiction with the other Hamilton equation - a minus sign is missing. Group velocities in agreement with the Hamilton equations are obtained only when in the time dependent phase of a quantum particle wave packet, instead of the Hamiltonian coming from the conventional Schrodinger equation, the Lagrangian is considered. This suggests us to consider the relativistic Lagrangian in the time dependent phase. In this way, the conventional relativistic principle of invariance of the time-space interval gets the more physically understandable form of the invariance of the time dependent phase of a quantum particle - the time dependent phase of a quantum particle is the same in any system of coordinates. Based on the relativistic time dependent phase invariance of a quantum particle, from the group velocities of this particle the relativistic kinematics and dynamics are obtained. The interaction with an electromagnetic field is described by a modification of the time dependent phase with a scalar potential conjugated to time and a vector potential conjugated to the space coordinates. In this paper, we find that, according to the formalism of the general relativity, any matter element in a field of forces is accelerated only perpendicularly to its velocity. This means that the matter propagation of a quantum particle can be conceived in planes perpendicular to velocity, while the mater distribution can be considered in a Fourier representation - quantum waves. We study the quantum particle dynamics in gravitational waves and derive the graviton spin. We consider a quantum particle in electromagnetic field and obtain a Schrödinger-Dirac-type equation, with additional terms describing the velocity dependence of the particle dynamics.
Keywords: Quantum particle; Wave packet; Group velocity; Lagrangian; Hamiltonian; Maxwell equations; Spin; Metric tensor; Covariant derivation; Gravitational wave; Graviton spin
IntroductionWe live in a period when the fundamentals of physics are put
under discussion [1-9]. For instance, in the framework of open quantum physics [10-14], we found that the general principle 2 of thermodynamics, of the entropy increase, was in fact not so general, but only for molecular systems. In the more general case of a system including a coherent electromagnetic field, the entropy could also decrease [12]. On this basis, an interesting phenomenon of environmental heat conversion into usable energy has been discovered [11-18]. Some time ago, we also discovered that the conventional Schrödinger equation is contradictory to the Hamilton equations [19], which are fundamental for the matter dynamics. Really, the dynamics of a particle is perceived in a three-dimensional space of coordinates defined by a position vector
1 1 1x y zr x y z= + +
, and in time, defined by a scalar t. For a matter object we define an inertial property called mass, 0M , and a dynamic quantity as the product of the mass with the velocity d
d
rr
t=
, called momentum, 0p M r=
. The dynamic law of a particle can be defined as a conservative process, by a conservative quantity, called energy,
( ) ( ) ( )0 ,E H r p T p U r= = + (1)
as a sum of the kinetic energy ( )T p , which depends on momen-tum, and the potential energy ( )U r , which depends on the coordi-nates. From the conservation condition
0 0d d d0
d d d
H HEr p
t r t p t
∂ ∂= + =
∂ ∂
(2)
we obtain the dynamic equations called Hamilton equations,
( )0d
d
Hr T p
t p p
∂ ∂= =
∂ ∂
( )0d
d
Hp U r
t r r
∂ ∂= − = −
∂ ∂
(3)
depending on the potential energy gradient called force, while the energy as a function of coordinates and momentum,
( ) ( )2
00
,2
pH r p U r
M= +
(4)
is called Hamiltonian. However, this classical description tells us nothing about the structure of the physical world. Only Quantum Mechanics tells us something about the structure of this world. Namely, that this world is composed of species of identical quantum particles. On one hand, experimentally, it has been found
How to cite this article: Eliade Stefanescu. Quantum Mechanics as a Fourier Representation of a Relativistic Quantum Particle . Ann Rev Resear. 2018; 4(4): 555641.0088
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that these particles are of a wavy nature. On the other hand, one could find that the simplest way to define a quantum particle is by a wave packet, with the momentum conjugated to the coordinates, and the energy conjugated to time, with a single quantum constant h [20-22]:
( )( )
( ) ( )i3
03/2
1, , d
2
pr Et
E r t p t e pψ ϕπ
−= ∫
( )
( )( ) ( )i
30 3/2
1, , d
2
pr Et
Ep t r t e rϕ ψπ
− −= ∫
(5)
In this case, one can define a momentum operator
ipr
∂= −
∂
(6)
and a Hamiltonian operator,
( )2 2
0 20
i2
H U r Et M r
∂ ∂= = − + =
∂ ∂
(7)
With these operators one obtains eigenvalue equations, for momentum,
( ) ( )i , ,E Er t p r trψ ψ
∂− =
∂
(8)
and for energy, as a Schrödinger equation:
( ) ( ) ( ) ( )2 2
20
i , , ,2
E E Er t U r r t E r tt M rψ ψ ψ
∂ ∂= − + =
∂ ∂
(9)
.
However, when the group velocities are calculated for the wave packets (5), which with (1) are of the form
( )( )
( ) ( ) ( ){ }i3
03/2
1, , d
2
pr T p U r t
E r t p t e pψ ϕπ
− + = ∫
( )( )
( ) ( ) ( ){ }i3
03 / 2
1, , d
2
pr T p U r tp t r t e rEϕ ψ
π
− − + = ∫
(10)
we obtain an erroneous equation, contradictory to the corresponding Hamilton equation:
( )d 0 - OKd
Hr T p
t p p
∂ ∂= =
∂ ∂
( )d Erroneous equation0 - d minus sign is missing
Hp U r
t r r a
∂ ∂= =
∂ ∂
(11)
We get back the minus sign only if instead the Hamiltonian
( ) ( ) ( ),oH r p T p U r= +
we consider the Lagrangian
( ) ( ) ( ) ( ) ( )2
r, r pr - H ,2
o o
M rOL p r T p U r U r= = − = −
(12)
In his case, the wave packets (10) take a form
( )( )
( ) ( ) ( ){ } 33 2
1, ,
2
ipr T p U r t
o or t p t e d pψ φπ
− − = ∫
( )( )
( ) ( ) ( ){ } 33 2
1, ,
2
ipr T p U r t
o Ep t r t e d rφ ψ
π
− − − = ∫
(13)
with group velocities in agreement with the Hamilton equa-tions (3):
( ) oHdr T p
dt p p
∂∂= =∂ ∂
( ) oHdp U r
dt r r
∂∂= = −∂ ∂
(14)
However, this description is still unrealistic, having an infinite spectrum of waves, as a function of the velocity r . A finite spectrum is obtained when the relativistic Lagrangian
( )O O o
rL r dt M C dt M cds
c= − − = −
(15)
is considered in the time dependent phase of a particle wave [13, 19-22]:
( )( )
( ) ( ), 3 33 2
1, ,
2
iMrr L r r toh
o or t r t e M d rψ φπ
− = ∫
( )( )
( ) ( ) , 33 2
1, ,
2
iMrr L r r toh
o r t r t e d rφ ψπ
− − = ∫
(16)
From these expressions with (15), we noticed that the invariance of the time-space interval of the Theory of Relativity is equivalent to the invariance of the time dependent phase of a quantum particle. We defined a Relativistic Quantum Principle of invariance of the time-dependent phase [23-27]. On this basis, we obtained the relativistic transform of the space-time coordinates, and the relativistic dynamics for the particle waves. We considered a quantum particle in a field described by a vector potential conjugated to coordinates, which is in agreement with the theory of Aharonov and Bohm [28], and a scalar potential conjugated to time. We obtained the Lagrange equation as a group velocity of the particle waves, the Lorentz force, the Maxwell equations, the relativistic transform of the electromagnetic field, and the spin-statistic relation [13, 19, 23-27].
In this paper, we adopt the converse starting point, not of the quantum mechanics, but of the general theory of relativity, where a quantum particle is a piece of matter with a density normalized to mass. In section 2, we find that, according to the general theory of relativity, this matter propagates in planes perpendicular to the velocity, which means that the matter propagation can be considered in a Fourier representation, as a group of waves. In section 3, we find a relativistic invariant of the matter density, and derive the equation of the matter conservation. In section 4, we find a dynamic equation of the velocity field as a function of the metric tensor. In section 5, we obtain the particle oscillation in a gravitational wave. In section 6, we find the particle acceleration in a gravitational wave, and define a gravitational invariant including a proper dynamic, called spin. In section 7 we consider the graviton spin as a matter rotation and find the angular momentum St=2. In section 8, we obtain a pseudo-energy tensor proportional to the square of the gravitational amplitude. In section 9, we
How to cite this article: Eliade Stefanescu. Quantum Mechanics as a Fourier Representation of a Relativistic Quantum Particle . Ann Rev Resear. 2018; 4(4): 555641.0089
Annals of Reviews and Research
obtain a fully relativistic dynamic equation. In section 10 we give a summary.
Quantum Particle as a Distribution of MatterWe consider a quantum particle as a distribution of matter
with the density as a positively defined quantity:
( ) ( ) 2, , , \ , , , |ox y z t M x y z tρ ψ= (17)
satisfying a normalization condition
( ) ( )1 2, , , | , , , | 1OM x y z t dxdydz x y z t dxdydzρ ψ− = =∫ ∫ (18)
According to the general theory of relativity [29, 30] in a system of curvilinear coordinates
( ) ( ), , 0,1, 2, 3; 1, 2, 3o jx x x jα α= = = (19)
we define a velocity field in the system of the proper time ,t s ct= ,
j jj jdx dx
v c cxdt ds
= = =
(20)
We define the differential time-space interval ds , which depends on the metric tensor v
gµ ,
2 vvds g dx dxµ
µ= (21)
From this expression, we obtain the fundamental property of the velocity field,
1.vvg x xµ
µ = (22)
Since the covariant derivative of the metric tensor is null, this property takes the form
: 0vx xµµ = : 0v
vx x xµµ⇒ = (23)
which means that any covariant variation of a velocity, or any covariant acceleration, is perpendicular to this velocity. At the same time, for any quasi free particle, i.e. when no external force is present, any matter element follows a geodesic trajectory,
, v vv v
dxx x x x
ds
µµ µ σ
σ= = −Γ
(24)
which means that the covariant acceleration is null:
: 0vvx xµ = (25)
In this case the acceleration , vvx xµ
of any differential element of matter includes only a component induced by the gravitational field described by the second kind Christoffel symbol
( ), , ,
1.
2v v v vv
g g g g gµ µλ µλσ λ σ λ σ λσ σ λΓ = Γ = + − (26)
When an external force is present, an additional, external acceleration Aµ arise,
, vvx x x x Aµ µ ν σ µ
νσ= −Γ +
(27)
This acceleration is of the form
( ) :,A x x x x xµ µ µ σ ν µ ννν νσ= + Γ =
(28)
which means that any acceleration induced by an external field is a covariant acceleration, while the acceleration induced by an internal, gravitational field, does not modify the covariant
acceleration. It is interesting that by multiplying the external acceleration (11) with the velocity xµ , and taking into account the fundamental property (7),
00
: == νν
µµ
µµ xxxAx
(29)
we find that any external acceleration is perpendicular to the particle velocity. This means that a particle can be conceived as a distribution of matter moving in planes perpendicular to the velocity (Figure 1) the external forces acting on the particle modifies the matter distribution in these planes, but not in between these planes (Figure 2).
Figure 1: Quantum particle accelerated in an external, attraction field, in planes, perpendicular to the velocity.
Figure 2: Quantum particle accelerated in a central field.
Thus, according to the general theory of relativity, the distri-bution of matter in a central field takes the form of an atom. This means that the distribution of matter of a quantum particle in a system of curvilinear coordinates can be considered in a Fourier representation,
( )( )
( ) ( )[ ] ( )( )
( )( )
( ) ( )[ ] ( )( ) ,ddd
,,,,,
21,
ddd,,,,
,2
1,
321321
,ii
2/3
321321
,i
2/3i
j
j
∫
∫
∂
∂=
∂
∂=
−−
−
xxxxxxzyxetxtP
PPPPPPPPP
etPtx
txxLxPj
zyxtxxLxPj
j
j
αα
αα
ψπ
φ
φπ
ψ
(30)
with the normalization condition (18),
( ) ( )( ) ( ) ( )
( ) 1ddd,,,,,ddd
,,,,, 321
321
2321321
10 =
∂∂
=∂∂
∫∫− xxxxxxzyxtxxxx
xxxzyxtxM ii ψρ (31)
for the matter density
( ) ( )20 ,, txMtx ii ψρ = (32)
How to cite this article: Eliade Stefanescu. Quantum Mechanics as a Fourier Representation of a Relativistic Quantum Particle . Ann Rev Resear. 2018; 4(4): 555641.0090
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and the Lagrangian
( ) βααβ
αα xxgcMxxL
20, −= (33)
while t is the proper time. In a stationary state, 0,0 00, == iggαβ , with the general property (6), the momentum is
.21
2
0
2000
0
20000
iij
jiij
iij
jiijjjj
xcgM
xxgxg
xcgM
xxgxgx
cMxcLP
−=
+−=
+∂
∂−=
∂
∂=
(34)
With these expressions, from the group velocity in the coordinate space, we find an equation, which with (6) takes the form an identity,
,
21d
d
jj
iij
jjj
xcxxg
xc
xg
xxgc
PLxcx
t
==
∂
∂
=∂
∂==
βααβ
βααβ
(35)
while from the group velocity in the momentum space, we find the Lagrange equation,
( ) .,dd
dd µµ xxL
xPcP
txcL
t jjjj
∂∂
===∂∂
(36)
For the Hamiltonian
( )txxLxcPH jjjj ,, −=
(37)
from its differential with the Lagrange equation,
,ddd
ddddd
ddddd
tt
HxxHP
PH
ttLxcPxPcxcPPxc
tLx
xLx
xLxcPPxcH
jjj
j
jj
jj
jjj
j
jj
jj
jjj
j
∂∂
+∂∂
+∂∂
=
∂∂
−−−+=
∂∂
−∂∂
−∂∂
−+=
(38)
we obtain the Hamilton equations
.
1
1
tL
tH
xH
cP
PH
cx
jj
j
j
∂∂
−=∂∂
∂
∂−=
∂∂
=
(39)
In this way, the dynamics of a quantum particle as a distribution of matter, of density (17) and the normalization condition (18), or of the wave functions (30) with the normalization conditions (31), is described by the dynamics of the matter differential elements according to the Hamilton equations (39).
The Matter ConservationWe consider the density normalization in two systems of
coordinates,
( ) ( ) 1dddd dddd 32103210 == ∫∫ ′′′′′ xxxxJxxxxxx µµ ρρ (40)
which depends on the Jacobian
( )µα′= ,Det xJ (41)
It is interesting that this Jacobian has the same elements as the tensor transformation. From the transform of the metric tensor between these two systems of coordinates,
βανβ
µααβ ′′
′′= gxxg ,, (42)
with the determinant
( )αβgg Det= (43)
we obtain the Jacobian
gg
J′−
−=
(44)
With the volume differential elements
32103210 dddddddd xxxxJxxxx =′′′′ (45)
the density integral on an arbitrary volume V takes the form( ) ( )∫∫ −=′− ′′′′′
VV
xxxxgxxxxxgx 32103210 dddddddd µµ ρρ (46)
which means an invariance relation of the matter density, depending on the quantity g− we call scalar density,
( ) ( ) gxgx −=′−′ µµ ρρ (47)
We define the matter flow four-vector
µµ ρxJ = (48)
with the conservation relation
0,,: =Γ+=Γ+= νµνµ
νν
νµνµ
µµµ
µ JJJJJ (49)
With the second kind Christoffel symbol
( )λσννλσσλνµλ
λνσµλµ
νσ ,,,21 ggggg −+=Γ=Γ (50)
and the symmetry of the metric tensor, the coefficient of the last term of equation (49) is
( )
( ) ( )( )
.21
21
21
21
,,
1,
1
,,,,
g
ggggg
gggggg
−
−=−−==
=−+=Γ
−− ννν
νλµµλ
λµννλµµλνµλµ
νµ
(51)
In this way, the conservation relation (49) of the null covariant divergence of the matter flow four-vector takes the form of a null ordinary divergence,
( ) 0,: =−=− νν
µµ gJgJ (52)
By integrating in a space volume,( ) 0d3
, =−∫V
xgJ µµ
(53)
and separating the time term from the space terms, we obtain the conservation relation under the more understandable form of the time variation of the matter contained in a volume V , due to a flow ( ) mim xtxJ ,ρ= through the surface of this volume,
How to cite this article: Eliade Stefanescu. Quantum Mechanics as a Fourier Representation of a Relativistic Quantum Particle . Ann Rev Resear. 2018; 4(4): 555641.0091
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( ) ( ) ( )∫∫∫Σ
−−=−−=
−
V
mmi
Vm
m
V
i xgxtxxgJxgxtx 23,
0,
30 d,dd, ρρ (54)
For a low velocity, 0 1x = , weak gravitational field, 1g = − , we obtain the ordinary conservation relation:
∫∫Σ
−=
V
rJrV
2
0,
3 ddρ (55)
Dynamic Equation of a Quantum Particle as a Distri-bution of Matter
As we showed in the previous sections, a quantum particle in a system of curvilinear coordinates can be considered as a distribution of matter with the positively defined density (32), as a function of the Fourier expansions (30), with the Lagrangian (33), and the canonical momentum (34). The matter density satisfies the matter conservation relation (55), which is
( )( ) ( )0 3 2
,0, d , di i m
V V
x t g x x x t g x xmρ ρΣ
− = − −∫ ∫
(56)
and the invariance condition (47), which is
( ) ( ), ,i iDI x t g x t gρ ρ ′ ′ ′= − = − (57)
From the second equation (30) with the fundamental property (21), the group velocity in the momentum space is
2 20 0 ,
d 1,
d 22j jj
x x gL jP M c M c g x x
t x g x x
α β
α βαβα β
αβ
αβ∂= = − = −∂
(58)
while from the derivative of (34) we obtain2 2
0 , 0
d 1 1
d 2 2k i i
j j ij k ijP cP M c g x x M c g xt
= = − −
(59)
We notice that the second derivative of the coordinate in this equation can be eliminated with the geodesic equation
x x xµ µ α βαβ= −Γ
(60)
From (58)- (60), we obtain
, ,i k i
ij k ij jg x x g x x g x xα β α βαβ αβ− Γ = (61)
With the expression (50) of the second kind Christoffel symbol, we obtain the dynamic equation
( ), , , ,
10
2i k
ij k j j jg x x g g g x xα βα β β α αβ− + + =
(62)
for the differential elements of the matter of a quantum particle.
Quantum Particle Oscillation in a Gravitational WaveA gravitational wave is a solution of the second order wave
equation
, 0g gµνρσ µν = (63)
We consider such a solution of the form
g u l x xµ νρσ ρσ µν= (64)
where ρσu is called amplitude tensor, while µνl is called polarization tensor. With this solution, from the wave equation (63) we obtain a normalization condition for the polarization
tensor 0 1 2 30 1 2 3 0g l l l l l lµν ν
µν ν= = + + + = (65)
We notice that the metric tensor, being symmetric, it can be diagonalized. By a space-time diagonalization, 00 =jg the dynamic equation (62) takes the form
( ) 0 0, , , 00,
1 1
2 2i k
ji k jk i ik j jg g g x x g x x− − + + = (66)
With the Christoffel symbol of the first kind,
( ), , ,
1
2kij ki j kj i ij kg g gΓ = + − (67)
this equation takes the form
0 000,
1
2i k
kij jx x g x xΓ = −
(68)
Since the gravitational waves are created by motions of large masses, we consider small velocities, 10 =x , which means that the dynamic equation (68) takes the simpler form
jki
kij gxx ,0021
−=Γ
(69)
By multiplying this equation with jx , and taking the first kind Christoffel symbol as a function of the Christoffel symbol of the second kind,
00,
1
2k l i j j
kl ij jg x x x g xΓ = − (70)
with the geodesic equation (60), we obtain a dynamic equation for the acceleration of a matter differential element in a gravitational potential 00g ,
0021 gxx l
l = (71)
By taking into account the derivative of the gravitational potential with the proper time,
αα xggxggxx j
jjj
ii ,00,0000 2
121
21
=== (72)
and a space-time diagonalization of the metric tensor, 0=αjg , we obtain the dynamic equation 0
21
,00 =
− j
jiii ggxx (73)
which is
021
,00 =− jjii ggx (74)
It is interesting that the first order derivatives of the gravitational potential (64) which is of the second order in the spatial coordinate, are of the first order in these coordinates. We obtain the dynamic equation
021
21
0000 =−=− ρρ
ρρ xluxxlugx ii
jjii
(75)
which with the condition
iil ρρ δ−= (76)
takes the form of a harmonic oscillation ii xux 002
1−= (77)
Let us consider a wave oscillating in the direction 1x , corresponding to a polarization tensor
0 1 2 30 1 2 31, 1, 0l l l l= = − = = (78)
which satisfies the condition (65). With a metric tensor for a weak gravitational field,
How to cite this article: Eliade Stefanescu. Quantum Mechanics as a Fourier Representation of a Relativistic Quantum Particle . Ann Rev Resear. 2018; 4(4): 555641.0092
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00 11 22 331, 1, 1, 1g g g g= = − = − = − (79)
the polarization tensor is
0,0,1,1 33221100 ==== llll (80)
For a system with rotational symmetry, while the amplitude tensor is of the form
oru ug u ugµν µνµν µν= = (81)
and a low velocity, while the fundamental property (21) is
001 g= (82)
we obtain uugu == 0000 (83)
From
uugu
ugguguug
4==
===νν
νν
νσ
µνσµ
νσ
µνσµ
(84)
we obtain the coefficient ννuu
41
= (85)
we call gravitational wave amplitude. With this expression, the dynamic equation is
1121 uxx −= (86)
which describes a harmonic oscillator with the frequency
/ 2c uω = (87)
Quantum Particle Acceleration in a Gravitational Wave and Proper Dynamics
In the previous section, we showed that a gravitational wave described by the wave equation (63), with a second-order solution (64), means an oscillation of any matter differential element with the frequency of this wave. However, we notice that the wave equation (63) accepts also a first-order solution,
σσµνµν xlug = (88)
with a normalization condition similar to
0== σσ
σρρσ llllg (89)
With this solution, from (74) we obtain the dynamic equation
021
00 =− jjii lugx (90)
which, with (5.20) is
021
=− ii ulx (91)
We notice that a solution of the form (88) means a constant acceleration of any differential element of matter in the direction of polarization of the gravitational wave. No oscillation is perceived by such an element in a gravitational wave. With the expression (81),
νσσµ
µννσ
µνσµ ugguguug === (92)
by multiplying with νl , we find an equation for amplitude tensor:
σννσ ullu = (93)
We consider a gravitational wave polarized in the direction 3x ,
1,0,1 3210 −==== llll (94)
and a weak gravitational field, with the metric tensor elements
1,1 3333
2222
1111
0000 −======== gggggggg (95)
With (93), from (92) we obtain the equations
,
0
0
33
03
32
02
31
01
30
00
uuu
uu
uu
uuu
−=−
=−
=−
=− (96)
which, with (95) become
.
0
0
33033333
0300
32023233
0200
31013133
0100
30003033
0000
uuuugug
uuugug
uuugug
uuuugug
−=+=−
=+=−
=+=−
=+=− (97)
From the first and the fourth equation (95) we obtain
uuuuuu 2,2 3300033300 =−−=+ (98)
From the second equation (98) with (84)
( ) ( )332211003333
2222
1111
0000
3300 21
21
212 uuuuuguguguguuuu −−−=+++===− ν
ν
we obtain
uuu 22211 −=+ (99)
With the contravariant elements of the polarization tensor
αβββαα
µνβναµαβ ugguggu == (100)
we calculate the gravitational invariant
( )
( ) ( )
( ) ( )[ ]( ) .22
21
2212
21
212
21222222
2
2212
22211
212
22211
22211
212
222
211
233003300
233
200
233
200
212
222
211
23300
231
223
212
203
202
201
233
222
211
200
2
uuuu
uuuuuuuu
uuuuuuuuuuu
uuuuuuuu
uuuuuuuIu
++−=
+++−=++=
−−−+−++++=
−−+++−−−
+++=−= αβαβ
(101)
The Graviton SpinAs we have shown in the preceding sections, a gravitational
wave is described by an amplitude tensor µνu We define the operator αδ
R of rotation with a differential angle αδ
in a plane ( )yx, . For a vector
( ) ( ) ( )[ ]rArArA 21 ,= ,
( ) ( ) ( ) ( ) ( ) ( )rAr
rrArAr
rrArrArAR
111111∂∂
×⋅+=∂∂
⋅×+=×+= αδαδαδαδ (102)
With the angular momentum operator
rrS
∂∂
×−= i (103)
the rotation operator (109) takes the form
( ) ( ) ( ) ( )rAerASrArAR S
1i111 i αδαδ αδ =+= (104)
How to cite this article: Eliade Stefanescu. Quantum Mechanics as a Fourier Representation of a Relativistic Quantum Particle . Ann Rev Resear. 2018; 4(4): 555641.0093
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This means that the operator of rotation with an arbitrary angle α is
αα
SeR i=
(105)
We notice that a rotation with an angle is equivalent to an inversion,
( ) ( ) ( )rArAerAR 11iS1 −== ππ (106)
This means that a vector ( )rA describes a rotation with the spin 1=S . For a rotation with an
Figure 3: Rotation of a vector A with an angle 2/π− in a plane ( )yx, .
Angle 2/π− , as it is shown in (Figure 3), we obtain
.or, 222
112
1221 AARAARARAARA −=−=⇒−== (107)
Now, let us consider a rotation with an angle α
in the plane ( )yx , of a tensor ( )yxu ,µν . From the invariance of an arbitrary scalar of the form
( ) ( ) ( ) ( )[ ] ( )[ ] ( )[ ]( ) ( )[ ] ( )[ ]( ) ( )[ ] ( )[ ] ,,,,
,,,
,,,,,,
iii
iii
yxBeyxAeyxue
yxBeyxAeyxue
yxBRyxARyxuRyxByxAyxuS
t
t
S
SSS
ναµαµν
α
ναµαµν
α
ναµαµν
ανµµν
=
=
==
(108)
we deduce the spin eigenvalue of a tensor 2−=tS , i.e. a spin 2=tS . For a rotation with an angle 2/π− ,
( )( )( ) ( )( )( ) ( )( )( ) ( )( )( )( ) ( ) ( ) ( ) .11
2221
2112
1222
11
2222
1221
2112
1111
2222
1221
2112
1111
BARuBARuBARuBARu
RBRARuRBRARuRBRARuRBRARu
BAuBAuBAuBAuS
+−−=
+++=
+++=
(109)
We obtain
.,,, 1221211211222211 uRuuRuuRuuRu −=−=== (110)
Let us consider the vectors
.222
1212
212
1111
AuAuAu
AuAuAu
+=
+=µ
µ
µµ (111)
By rotating this vector with an angle 2/π− ,
( ) ( )( ) ( )( ) ( ) ( )
( ) ( )( ) ( )( ) ( ) ( )
,212
1111
122
221
222
1212
222
1212
112
211
212
1111
AuAuAu
ARuARuRARuRARuAuR
AuAuAu
ARuARuRARuRARuAuR
−−=−=
−=+=
+==
−=+=
µµ
µµ
µµ
µµ
(112)
we obtain again
.,,, 1221112221122211 uRuuRuuRuuRu −==−== (113)
Let us consider a rotation of the scalar (114) with an angle 4/π− :
( )( )( ) ( )( )( ) ( )( )( ) ( )( )( )( )( )( ) ( )( )( )( )( )( ) ( )( )( )
( ) ( ) ( )[ ] ( ) ( ) ( )[ ]( ) ( ) ( )[ ] ( ) ( ) ( )[ ]
( ) ( ) ( )( ) ( ) ( )
−−++−+
−+
+=
++++
+++=
++
+=
+++=
+++=
−−−−
−−−−
−−−−
−−−−
122122
221121
112212
211211
22/i2222/i22
12/i2122/i21
22/i1212/i12
12/i1112/i11
24/i24/i22
14/i24/i21
24/i14/i12
14/i14/i11
2222
1221
2112
1111
2222
1221
2112
1111
21
21
21
21
21
21
21
21
BABARuBABARu
BABARuBABARu
BeABAeRuBeABAeRu
BeABAeRuBeABAeRu
BeAeRuBeAeRu
BeAeRuBeAeRu
RBRARuRBRARuRBRARuRBRARu
BAuBAuBAuBAuS
ππππ
ππππ
ππππ
ππππ
(114)
We obtain the relations:
( )
( )
( )
( ).212121
21
221121
221112
211222
211211
RuRuu
RuRuu
RuRuu
RuRuu
−=
−=
+=
+−=
(115)
With the relations
1i22i
42i
4i t
−==⇒===− π
πππ
eReeeRS (116)
equations (7.21) become:
( )
( )
( )
( ).2121
21
21
112221
112212
211222
211211
uuRu
uuRu
uuRu
uuRu
−=
−=
+−=
+= (117)
By rotations R , the two terms of the gravitational invariant (104) are transformed one another:
( )[ ] ( )
( ) ( )[ ] .2212
2221
222211
212
2212
22211
2
u
u
IuuuRRu
uRuuuRIR
=+−+=
++−=↓↓
(118)
This means that this invariant includes two components: a component u , which means a matter acceleration according to the expression (91), and an oscillating component 22211211 ,, uuuu = , describing an oscillatory motion - the spin.
Gravitational Action and Pseudo-Energy TensorWe consider the gravitational action as an integral of the total
curvature of the space,
∫= xRI g4d (119)
where we used the notation g−= . With the Ricci tensor (Appendix A),
αβα
βµν
αβν
βµα
ααµν
ανµα
αµναµν ΓΓ−ΓΓ+Γ−Γ== ,,RR (120)
the total curvature is of the form (Appendix B)
How to cite this article: Eliade Stefanescu. Quantum Mechanics as a Fourier Representation of a Relativistic Quantum Particle . Ann Rev Resear. 2018; 4(4): 555641.0094
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LRRgR −== ∗µν
µν (121)
Where
( )( ) .
,,αβν
βµα
αβα
βµν
µν
ααµν
ανµα
µν
ΓΓ−ΓΓ=
Γ−Γ=∗
gL
gR (122)
In the action (119) with these terms, (Appendix C)
( )∫ −= ∗ xLRI g4d (123)
we integrate the first term by parts,( ) ( ) ( ) ( ) α
µναµνα
µανµν
ααµν
µνν
αµα
µν Γ+Γ−Γ−Γ= ,,,,* ggggR
(124)
and, since the variations of the action at the integration limits are null by its definition, we eliminate the total derivatives. In this case, we can take
( ) ( ) αµαν
µναµνα
µν Γ−Γ= ,,* ggR (125)
With the formulas
,,
,
,,
σασα
σραρσααρσ
αρσνσµρ
αµν
Γ=
Γ+Γ=
−=
g
gggg (126)
the first derivative in (124) is
( )( )
( ) ,
,,,
σασ
µννρα
µρµσα
νσ
σασ
µνσραρσα
νσµρα
µνα
µνα
µν
Γ+Γ−Γ−=
Γ+Γ+Γ−=
+=
ggg
ggg
ggg (127)
while, by contracting the suffixes ν and α , we obtain the second term,
( ) ( )
,
µσν
νσ
σνσ
µννρν
µρµσν
νσν
µν
Γ−=
Γ+Γ−Γ−=
g
gggg (128)
With these expressions, the symmetry relation of the Christoffel symbol, and (122), we obtain the first term (124) of the action integral (123),
( )( )( )( )
L
gg
gg
ggg
ggggR
2
22
22
2
*
=
ΓΓ−ΓΓ=
ΓΓ−ΓΓ=
ΓΓ+ΓΓ+Γ−=
ΓΓ+ΓΓ+Γ−Γ−=
ασν
σµα
νµσασ
αµν
µν
αµν
µσα
νσσασ
αµν
µν
αµα
µσν
νσαµν
σασ
µνµσα
νσ
αµα
µσν
νσαµν
σασ
µννρα
µρµσα
νσ
(129)
With this expression, for the gravitational action we obtain the expression
( )∫ ∫∫
∫∫∫==
=−== ∗
32104
444
ddddd
ddd
xxxxx
xLxLRxRI g
LL
(130)
with the gravitational Lagrangian density
( ) µνβαν
αµβ
βαβ
αµν gL ΓΓ−ΓΓ==L (131)
which depends on the variables µνg which here we consider as “coordinates” and the variables σµν ,g considered as “velocities”. With these “coordinates” and “velocities”, we obtain the variation
of the gravitational action
,0d
dd
4
,,
4,
,
4
=
∂∂
−∂∂
=
∂∂
+∂∂
==
∫
∫∫
xggg
xgg
gg
xI g
αβνναβαβ
ναβναβ
αβαβ
δ
δδδ
LL
LLL
(132)
which leads to the Lagrange equations,
αβνναβ gg ∂∂
=
∂∂ LL
,, (133)
with terms of the Lagrangian variation
ναβναβ
αβαβ
δδδ ,,
gg
gg ∂
∂+
∂∂
=LLL (134)
These equations suggest the definition of an energy pseudo-tensor
LL νµµαβ
ναβ
νµ gg
gt −
∂∂
= ,,
(135)
However, for the Lagrangian (131), from (B.7) we obtain
( )
( )αµν
βαβ
ανβ
βµαµν
σασ
νβ
ναβ
ναβ
ΓΓ−ΓΓ=∂
∂
Γ−Γ=∂
∂
g
gg
L
L
,
(136)
which suggest the definition of the energy pseudo-tensor
( )( ) , Lνµµ
αβσασ
νβ
ναβ
νµ gggt −Γ−Γ= (137)
It is interesting that this change of coordinates, mn Qq → ,
µµ ,, nn
mm q
qQQ∂∂
= (138)
does not modifies the terms of the equation (135),
.,,
,,
,,
,,
,
,,
,
,
,,
,
µν
µν
µνσ
σµ
ν
σ
σµ
ν
σ
σµ
ν
mm
nn
m
m
nn
m
mn
n
nn
m
mn
n
m
mn
n
qqQ
Q
qgqQ
q
qqQ
q
∂∂
=∂∂
∂∂
=
∂∂
∂∂
=∂∂∂
∂
∂∂
=∂∂
∂∂
=∂∂
LL
LLLL
(139)
This means that the two expressions (135) and (137) define the same energy pseudo-tensor. With the expressions (88) of the metric tensor, and the expression
( ) ( )
µαβ
µρσβσαρ
µρσβσαρ
µαβ
µµαβαβ
µαβαβ
µαβαβ
µµ
lulugggggg
ulluggggggg
−=−=−=
===−=−=
,,
,,,, 221
21
21
21
(140)
we obtain
( )( ) µ
αβµ
αβαβ
µαβ
µαβ
µαβ
µαβ
µαβ
luglugu
ulgluggg
=−−=
+−=+=
2
2,,,
(141)
At the same time,λαβ
νλναβ Γ=Γ g (142)
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With
( ) ( )λαβαλββλαλαβαλββλαλαβ lululuggg −+=−+=Γ21
21
,,, (143)
which is
( ) ( )( )ulglglg
lulululululug
ναβα
νββ
να
ναβα
νββ
ναλαβαλββλα
νλναβ
−+=
−+=−+=Γ
21
21
21
(144)
While
( )
( ) .2ln122
1
21
21
,,,
,1
,,,,
αααα
α
αλσσλ
λασαλσσλασλ
λασσλσ
ασ
ulgg
ggg
ggggggg
===−−
==
=−+=Γ=Γ
−
(145)
With the expressions (144), (145), and (141), the first term of the energy pseudo-tensor (137) is
( )( ) ( )( )
µν
µν
µν
µανα
µν
ανα
βνβ
µαβ
ανβ
ναβα
νββ
ναµ
αβσασ
νβ
ναβ
llullullu
lulglullglg
lugulgulglglggg
222
22
,
422
2421
221
−=−−=
−−+=
−−+=Γ−Γ
(146)
The second term of the pseudo-energy (137) is proportional to the Lagrangian
( ) µνβαν
αµβ
βαβ
αµν gL ΓΓ−ΓΓ==L
(147)
which, according to (C.7) and the normalization condition (89), is
02 2 =−= αα lluL (148)
With (146) and (148), the pseudo-energy tensor (137) is
µνν
µ llut 24−= (149)
We notice that, according to (89), the pseudo-energy satisfies the normalization condition of a null trace:
{ } 0Tr =νµt (150)
Non-zero elements arise only for a non-zero amplitude of a gravitational wave, as an acceleration field (91). When the amplitude is null, only a proper motion is described by the elements of the amplitude tensor in the invariant (118), which is the gravitational spin.
Relativistic Wave Equation of a Quantum Particle in Electromagnetic Field
In a flat space, corresponding to the special theory of relativity, the wave equation of a particle in electromagnetic field, the wave functions (30) take a simpler form
( )( )
( ) ( )[ ]
( )( )
( )( )[ ]
,d,2
1,
d,2
1,
3,,i
2/3
3,,i
2/3
∫
∫
−−
−
=
=
retrtP
PetPtr
ttrrLrP
ttrrLrP
ψπ
φ
φπ
ψ (151)
depending on time ctxt =0, , the Lagrangian
( ) ( ) ( ) treUrtrAetcMttrrLcr dd,d0d,, 2
22 1
−= +−−
(152)
and the momentum
( ) ( ) ( )trAeptrAerMtrrLr
P
cr
,,,,
2
21
0
+=+=∂∂
=−
(153)
From the group velocity in the momentum space, we obtain the Lagrange equation
( ) ( )trrLr
trrLrt
Pt
,,,,dd
dd
∂∂
=∂∂
= (154)
For an electromagnetic field with a time independent scalar potential as of an atom, and a time dependent vector potential as of a radiation field. It is interesting that, in this case, we obtain time the Hamiltonian as a function not explicitly depending on time, but only on coordinate velocity - the energy:
( ) ( )
( ) ( ) ( )
( ) ( ) .,
1
,1,
1
,,,,
2
2
20
2
22
0
2
2
20
rrEreU
cr
cM
reUrtrAecrcMrtrAe
cr
rM
trrLrPtrPH
=+
−
=
−+−−−+
−
=
−=
(155)
From (153), we obtain the velocity dependent term of the energy as a function of the canonic potential, which is the eigenvalue of the quantum momentum operator,
( )[ ] 220
2220
2
2
220
2
2
220 ,
11cMtrAePcM
crrM
crcM
+−=+−
=−
(156)
We obtain the relativistic Hamiltonian
( ) ( )[ ] ( )reUtrAePcMctrPH
+−+=222
0 ,,, (157)
while, with (155), the particle wav function (151) takes the form
( )( )
( ) ( )[ ]{ }( )
( ) ( )[ ]∫∫
−+−−== PetP
hPetP
htr
trPErPtrPHrPrP
3i
2/33,i
2/3 d,2
1d,2
1, φπ
φπ
ψ
(158)
We obtain the relativistic wave equation
( ) ( )[ ] ( ),,,,i trrPrPHtrt
ψψ −=∂∂
−
(159)
which depends on the momentum operator
( )r
trAepP
∂∂
−=+= i, (160)
With the total time derivative
rPtr
rtt
idd
+∂∂
=∂∂
+∂∂
=
(161)
How to cite this article: Eliade Stefanescu. Quantum Mechanics as a Fourier Representation of a Relativistic Quantum Particle . Ann Rev Resear. 2018; 4(4): 555641.0096
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we obtain the relativistic wave equation of the form of the Schrödinger equation, but with the minus total time derivative instead of the partial time derivative of the wave function, and the relativistic Hamiltonian:
With relativistic Hamiltonian:
( ) ( ) ( ) ( ) ( )trrrEtrrPHtrt
,,,,,ddi
ψψψ ==− (162)
With relativistic Hamiltonian
( ) ( ) ( ) ( )reUpppcMcreUpcMcrpH
++++=++= 33221100222
0, αααα
(163)
depending on Dirac’s operators
{ } ijji δαασ
σα
σσ
ασ
σαα 2,,
00
,0
0,
00
,1̂0
01̂3
33
2
22
1
110 =
=
=
=
−=
(164)
as functions of the Pauli operators
{ } ijji δσσσσσ 2,,10
01,
0ii0
,0110
321 =
−
=
−=
=
(165)
the relativistic dynamic equation (162) takes the form
( ) ( ) ( )[ ] ( ) ( )trEtrreUpcMctrt
,,,ddi 00
ψψααψ =++=− (166)
depending on the mechanical momentum
( )trAer
p ,i
−∂∂
−= (167)
and Dirac’s spin operator
( )
==
00
,, 321 σσ
αααα
(168)
as a function of the Pauli spin operator
( )321 ,, σσσσ = (169)
With these operators, the particle wave function is split into four components:
=
4
3
2
1
φφφφ
ψ
(170)
For these components, from (166) we obtain the explicit system of equations
( ) ( )( ) ( )
( ) ( ) ( ) ( ) ( ) ( )
( ) ( )( ) ( )
21 0 1
4 4 3
22 0 2
i , ,
i , , ic i , , i , ,
i , ,
i
x y z
x y z r t M c eU r r tt x y z
c eA r t r t eA r t r t c eA r t r tx y z
x y z r t M c eU r r tt x y z
c
ϕ ϕ
ϕ ϕ ϕ
ϕ ϕ
∂ ∂ ∂ ∂− + + + = + ∂ ∂ ∂ ∂ ∂ ∂ ∂ + − − − − − + − − ∂ ∂ ∂
∂ ∂ ∂ ∂− + + + = + ∂ ∂ ∂ ∂
+ −
( ) ( ) ( ) ( ) ( ) ( )
( ) ( )( ) ( )
( ) ( ) ( ) ( )
3 3 4
23 0 3
2 2
, , i i , , i , ,
i , ,
i , , i i , ,
x y z
x y
eA r t r t c eA r t r t c eA r t r tx y z
x y z r t M c eU r r tt x y z
c eA r t r t c eA r t r t cx y
ϕ ϕ ϕ
ϕ ϕ
ϕ ϕ
∂ ∂ ∂ − + − − − − − ∂ ∂ ∂
∂ ∂ ∂ ∂− + + + = − + ∂ ∂ ∂ ∂
∂ ∂ + − − − − − + ∂ ∂
( ) ( )
( ) ( )( ) ( )
( ) ( ) ( ) ( ) ( ) ( )
1
24 0 3
1 1 2
i , ,
i , ,
i , , i i , , i , , ,
z
x y z
eA r t r tz
x y z r t M c eU r r tt x y z
c eA r t r t c eA r t r t c eA r t r tx y z
ϕ
ϕ ϕ
ϕ ϕ ϕ
∂ − − ∂ ∂ ∂ ∂ ∂− + + + = − + ∂ ∂ ∂ ∂ ∂ ∂ ∂ + − − + − − − − − ∂ ∂ ∂
(171)
depending on the rest energy, velocity, and the electromagnetic potentials.
SummaryWe considered a quantum particle with a mass 0M , as a
continuous distribution of matter in a curvilinear system of coordinates, with a density positively defined as the square of the absolute value of a complex function. As a description of the matter dynamics, we defined a velocity field and showed that, according to the general theory of relativity, any covariant acceleration of a matter differential element, under the action of an external field, is perpendicular to this velocity. This means that the matter moves in planes perpendicular to the velocity, and that, for the complex
function one can consider a Fourier series expansion in waves - quantum mechanics.
We considered the matter conservation, and obtain a dynamic equation for velocities, depending on the metric tensor elements. When, in the equation of a gravitational wave, we consider a metric tensor of the second degree in coordinates, from the dynamic equation we obtain a harmonic oscillation of a differential element of matter. In this case, the metric tensor is proportional to an amplitude tensor, and a polarization tensor which satisfies a normalization condition. When we consider a metric tensor of the first degree in coordinates, we obtain an acceleration of a matter differential element. In this case, the metric tensor is
How to cite this article: Eliade Stefanescu. Quantum Mechanics as a Fourier Representation of a Relativistic Quantum Particle . Ann Rev Resear. 2018; 4(4): 555641.0097
Annals of Reviews and Research
also proportional to an amplitude tensor, but with a polarization vector, which also satisfies a normalization condition. For an amplitude tensor, we defined a scalar we called gravitational wave amplitude. In both cases, of a first-order solution, or of a second-order solution, the acceleration is proportional to this amplitude. For the amplitude tensor of a first-order solution of the metric tensor, we obtained a gravitational invariant, as a sum of the square of the gravitational amplitude plus two terms which, by rotation, transform one another. When the gravitational wave amplitude is null, no acceleration is present, while the two terms, describe an inner motion of the matter, we call spin. We obtain the gravitational spin 2 as an eigenvalue of the rotational angular momentum.
We considered the gravitational action as a time-space integral of the total curvature with the scalar density and obtained a Lagrangian depending on the elements of the metric tensor, and the derivatives of these elements with the contravariant coordinates. We defined a pseudo-tensor of energy, proportional to the square of the gravitational wave amplitude, which satisfies the normalization condition of a null trace.
In a flat space, for wave functions defined in a time-space system of coordinates, we obtained a relativistic wave equation of a quantum particle in electromagnetic field. Unlike the conventional Schrödinger-Dirac equation, instead of the partial time derivative of the wave function this equation contains the minus total time derivative, which includes an additional dependence on velocity. These differences appear from dependence of the time dependent phases of the wave functions on the Lagrangian, as the product of the momentum with the velocity minus the Hamiltonian, instead of the Hamiltonian in the conventional Schrödinger-Dirac equation. We obtained an explicit system of equations for the four components of the wave function, describing the velocity dependence of the particle dynamics.
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How to cite this article: Eliade Stefanescu. Quantum Mechanics as a Fourier Representation of a Relativistic Quantum Particle . Ann Rev Resear. 2018; 4(4): 555641.0098
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