The generalized quantum mechanics

of Einstein «deinterlaced» photon

and Casimir force

BEILINSON A. A.

Department of Theoretical Physics

Peoples’ Friendship University of Russia

Russia, 117198, Moscow, Miklukho-Maklaya st., 6

RUSSIA

Abstract: Based on Majorana equations the e.-m. field as initially quantum object having isomorphic represen-

tation as quantum field of «deinterlaced» photon is considered. The calculation of Casimir force magnitude in-

terpreted as consequence of an energy measurement of the generalized quantum field of a «deinterlaced» photon

in the state corresponding to a «Feynman path» element is given. A metallic mirrors here plays role of classic

apparatus measuring energy of this field.

Key-Words: generalized quantum mechanics, generalized path integral, Casimir force, Feynman paths, quantum

field of a photon.

Received: December 14, 2022. Revised: September 17, 2023. Accepted: October 15, 2023. Published: November 29, 2023.

1 Introduction

As known (see [10]), the Casimir forces are called the

forces of attraction that arise between neutral metallic

mirrors at a small distance among them. These forces

have the quantum-electrodynamic origin; despite the

smallness of them, their dependence on distance be-

tween mirrors (inverse proportionality of 4th degree)

has been experimentally determined. Here the e.-m.

field in the article is understood as a fundamentally

quantum object that has no classic limit.

Note. Majorana equations of e.-m. field (see [1])

not contain Planck constant has well as the usual

Maxwell equations. However the photon spin is not 1,

but is h, which is why there should be a multiplier h

in the l.h.s. and r.h.s. in the Majorana equations;

these equations turn out to be equivalent to the pre-

vious ones only when h= 0 (for h= 0 the Maxwell

equations just disappear).

Therefore the Maxwell equations describes a

quantum object not dependencies on quantity h= 0

and therefore that has no classic analogue, so that

all laws of classic e.-d. is interpreted as means of

such quantum field. This fundamental property the

quantumness of e.-m. field existing in two isomor-

phic forms representing physically radically different

quantum field (see [4]). One of them (the «deinter-

laced» photon field) is responsible for the origin of

Casimir force.

In the article, Casimir force is interpreted as result

of a energy macroscopic measurement of quantum e.-

m. field of the «deinterlaced» photon corresponding

to small element of «Feynman path» of the photon,

in field of which a mirrors are placed, that are such a

classic apparatus.

Since the considered problem is obviously one-

dimensionally the generalized Green function of ar-

bitrary e.-m. field in its usual and isomorphic forms

is constructed and studied.

In the article uses terminology and notation of

monographs [6, 7, 8]; in calculations, we assume the

light velocity c= 1.

2 Generalized Green function of

one-dimensional quantum e.-m.

field as functional on bump

functions

Recall that Maxwell equations of e.-m. field in Majo-

rana variables [1] for photon ξt(x) = Et(x) + iHt(x)

and antiphoton ¯

ξt(x) = Et(x)−iHt(x)are become

equations

i∂

∂t ξt(x) = (S, ˆp)ξt(x),

−i∂

∂t ¯

ξt(x) = (S, ˆp)¯

ξt(x),

(1)

where ˆp=1

i∇, and Sis photon spin opera-

tors (infinitesimal operators of rotation around co-

ordinate axes in representation corresponding to

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Beilinson A. A.

E-ISSN: 2732-9941

Volume 3, 2023

weight 1, see [5]).

S1=



0,0,0

0,0,−i

0, i, 0

, S2=



0,0, i

0,0,0

−i, 0,0

,

S3=



0,−i, 0

i, 0,0

0,0,0

.

ξt(x),¯

ξt(x)are supposed independent, see [1], that is

why the Green function of these equations is repre-

sented by direct product of 3×3matrices and their

solution is column of 6elements. Therefore, for the

momentum representation of these equations are as

follows:

i∂

∂t ˜

ξt(p) = (S, p)˜

ξt(p),

−i∂

∂t

ξt(p) = (S, p)¯

ξt(p).

We will consider a quantum e.-m. field correspond-

ing to states of a photon on the zaxis, in which the

coordinates x,yof photon is any (and hence the mo-

mentum px=py= 0 corresponding to them).

The problem on the quantum e.-m. field of an an-

tiphoton is solved similarly. Almost to the end of the

article will be investigated only a photon.

Such e.-m. fields in momentum representation, it

is easy to see, satisfies the Shrödinger equations

i∂

∂t ˜

ξ(1)

t(pz)

ξ(2)

t(pz)!=Szpz ˜

ξ(1)

t(pz)

ξ(2)

t(pz)!,

∂

∂t ˜

ξ(3)

t(pz) = 0,

the last component is «frozen» and does not partici-

pate in evolution.

It is clear that a generalized Green function of this

equation is

Ξt(pz) = expt0,−pz

pz,0.

Bearing in mind the constructing of the coor-

dinate representation the generalized Green func-

tion of this field as functional on columns of bump

functions φ(z)∈K(as Fourier preimage of so-

lution in the momentum representation as func-

tional R¯

Ξtψ1

ψ2dpzon columns of analytic func-

tions ψ(p)∈Z(см. [6]). Here

Z¯

Ξtφ1

φ2dz =1

2πZ¯

Ξtψ1

ψ2dpz

Let us consider the momentum representation of gen-

eralized Green function of considered e.-m. field of a

photon in details.

Remark that the momentum representation of the

solution contains in power coefficient a Hermit ma-

trix, therefore which can be reduced to diagonal view

by unitary transform ˜

Q(pz). Therefore we have

Ξt(pz) = exp−i0,−itpz

itpz,0=

=˜

Q+(pz)exp(−it|pz|),0

0,exp(it|pz|)˜

Q(pz),

(2)

where

Q(pz) = 1

√2−isgn pz,1

isgn pz,1.

But the Fourier preimages of numeric function-

als exp(it|pz|)and sgn pzon Zare the quantum

Cauchy functional (see [3, 2])

Cit(z) = 1

2(δ(t−z) + δ(t+z)) +

2π1

t−z+1

t+z(3)

and, accordingly, the func-

tional −i

πz (see [6], p. 360 formula 19) on bump

functions φ(z)∈K.

Note. Previously it was shown that quantum

Cauchy process (see [2]) is the correct analytic con-

tinuation in time of the Cauchy process transition

probability 1

π·t

t2+z2(see [9]) from the real semiaxis to

the imaginary axis. Remark that formula (3) can also

be obtained by a simple calculation using improper

integrals.

Hence, it easy to see, that

Ξt(pz) =

2 e−it|pz|+eit|pz|,sgn(pz)·2isin(t|pz|)

−sgn(pz)·2isin(t|pz|), eit|pz|+e−it|pz|!=

=cos(tpz),sin(tpz)

−sin(tpz),cos(tpz).

Hence, it easy to see, that we have coordi-

nate representation the generalized Green func-

tion R¯

Ξtφ1

φ2dz of Maxwell–Majorana equations,

where

Ξt(z) =

2δ(t−z) + δ(t+z), i (δ(t−z)−δ(t+z))

−i(δ(t−z)−δ(t+z)) , δ(t−z) + δ(t+z).

(4)

Thus the generalized Green function of the studied

e.-m. field is concentrated in points z=±t, which

are the wave front, describing the initial state evolu-

tion of the field concentrated in point z= 0.

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DOI: 10.37394/232020.2023.3.9

Beilinson A. A.

E-ISSN: 2732-9941

Volume 3, 2023

We shall return to relation (2) that has a physical

interpretation.

Indeed, the matrix diagonalization underlying this

equality is interpreted as cut-off the spin interaction

between components of the considered generalized e.-

m. field and the rise in result a new field with the mo-

mentum representation of the generalized Green func-

tion

Ξt(pz) = exp(−it|pz|),0

0,exp(it|pz|)

with «deinterlaced» components. Here the unitary

operator ˜

Q(pz)realizes both «deinterlacing» and «in-

terlacing» of the field components.

Moreover the coordinate representation the gener-

alized Green function of this field with «deinterlaced»

components is R¯

Ξtφ1

φ2dz (see (4)), where

Ξt(z) = C−it(z),0

0, Cit(z).(5)

Let us compare those generalized quantum e.-m.

fields with unitary equivalent Green functionals Ξt(z)

and \

Ξt(z). Since

R¯

Ξt(z)φ1(z)

φ2(z)dz =

2φ1(t) + φ1(−t) + i(φ2(t)−φ2(−t))

−i(φ1(t)−φ1(−t)) + φ2(t) + φ2(−t),

then this functional is concentrated in points z=±t

of a quantum e.-m. field wave front. At the same time

the functional

R¯

Ξt(z)φ1(z)

φ2(z)dz =

2



φ1(t)+φ1(−t)

2+i

2πv.p. Rφ1(z)

t−z+φ1(z)

t+zdz

φ2(t)+φ2(−t)

2−i

2πv.p. Rφ2(z)

t−z+φ2(z)

t+zdz



is concentrated on the exterior of wave front (outside)

in R(1), see 4rd section of present work.

Remark.So far, a bump functions have played

a role of coordinates, in which it turned out to be

possible to work with Green functionals of gen-

eralized quantum e.-m. fields. It turns out that

with help of complex bump functions ψ(z) =

φ(z) + iϕ(z)(φ(z), ϕ(z)∈K) we can construct a

Shrödinger equations corresponding to Green func-

tionals of quantum e.-m. fields and therefore their

physically interpreted solutions.

Indeed, the existence of integral

Z¯

Ξt(z−z0)ψ1(z0)

ψ2(z0)dz0=ψ1(z)

ψ2(z)

interpreting as solution of a Cauchy problem for a

Shrödinger type equation with Hamiltonian

H= 0, i ∂

∂z

−i∂

∂z ,0!

(Cf. (1)).

From this point on to simplify the recording it will

be considered one of two independent components of

the quantum e.-m. field with Shrödinger equation and

its physically interpreted solution

About similar procedure for \

Ξt(z)and Cit(z)

see 4rd section of present work.

Note. The passage from Ξt(z)to \

Ξt(z)by means

of unitary (and therefore isomorphic) transform is the

fundamental rebuilding of the quantum e.-m. field,

representing its a completely new face. If the solution

representation in the wave form (4) was known, then

the constructed isomorphic solution (5) has a diffuse

character that allowed to discover and construct the

generalized quantum measure in the space of a photon

«Feynman paths» with Hilbert instantaneous veloci-

ties (see [2]).

3 Generalized functional integral

corresponding to generalized

Green function of constructed

quantum e.-m. field of

«deinterlaced» photon

Recall that the generalized quantum Cauchy pro-

cess Cit(z)— see (4), (and therefore the \

Ξt(z)) is

continuable to a generalized countably additive com-

plex measure in the space dual the space of Hilbert

instantaneous velocities of a photon, that is turned

out to be a compact part of the continuous function

space, see [2, 8].

The countably additivity of the quantum gen-

eralized Cauchy measure yields to possibility of

quantum-theoretic expansion the state term of a pho-

ton to states on trajectories («Feynman paths»), con-

tinuous trajectories with Hilbert derivative.

We will assume that ∆t1, . . . , ∆tn(t0= 0,

tn=t) is certain partitioning of a time interval [0, t],

and ∆zj(j= 1, . . . , n) is shifts of the photon at the

appropriate time intervals.

Using the kernel theorem (see [8]) we have

R n

j=1

Ci∆tj(∆zj)!×

×φ(∆z1, . . . , ∆zn)dzj. . . dzn,

where φ(z1, . . . , zn)is bump functions of nvariables.

This allows to interpret that functional as a gen-

eralized state of the photon located sequentially on n

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DOI: 10.37394/232020.2023.3.9

Beilinson A. A.

E-ISSN: 2732-9941

Volume 3, 2023

segments ∆zjat the appropriate time intervals ∆tj.

This implies that the convolution of these states

R n

j=1

Ci∆tj(∆zj)!×

×φ(∆z1+. . . + ∆zn)dzj. . . dzn=

=R¯

Cit(z)φ(z)dz

is the state of the photon at the last moment of time.

The existence of the generalized Cauchy measure

gives possibility of passage to the limit in the writing

of the quantum Cauchy process and passage to the

generalized functional functional integral on «Feyn-

man paths»

limmax ∆tj→0R n

j=1

Ci∆tj(∆zj)!×

×φ(∆z1+. . . + ∆zn)dzj. . . dzn=

=R{zτ}t

τ=0

Cidτ (dz(τ))×

×φ[z(τ)] t

τ=0

dzτ=

=R¯

Cit(z)φ(z)dz,

where {zτ}is the support of generalized quantum

Cauchy measure — «Feynman paths» — set of con-

tinuous trajectories on [0, t], which are compact in

topology of uniform convergence, dzτis differential

at constant time, dz(τ) = ˙z(τ)dτ (˙z(τ)∈L2(0, t)),

and φ[z(τ)] are bump functionals on {zτ}.

Remarkably, those solution forms of the quantum

fields, belonging to isomorphic Hilbert spaces with

common scalar product, that allowed to discover the

existence of such measure with similar properties in

usual form of the quantum e.-m. field, on average (in

eikonal approximation) giving the equations of clas-

sic electrodynamics.

4 Physic attribution of quantum

e.-m. field of «deinterlaced» photon

and Casimir forces

We consider the structure of the quantum field of

«deinterlaced» photon with the generalized Green

function Cit(z)(see (3)) with the Hamilton func-

tional Rˆ

H(z)ψ0(z)dz =1

πRz−2ψ0(z)dz and inte-

gral Shrödinger equation

i∂

∂t ψt(z) = 1

πZα−2ψ0(z−α)dα.

with Cauchy problem solution on each small time in-

terval ∆t

ψ∆t(z) = ψ0(z) + i∆tRˆ

H(z−α)ψ0(α)dα−

−i∆tRˆ

H(α)ψ0(z−α)dα

Taking into account the definition of functional α−2

v.p. Rα−2ψ0(z−α)dα =

=limε→0 −ε

−∞

∞

ε!α−2ψ0(z−α)dα

where εis any (see [6], p. 52 formula 7), in our prob-

lem, where ε > 0, we have

v.p. Rα−2ψ0(z−α)dα =

= −ε

−∞

∞

ε!α−2ψ0(z−α)dαε→0

where different from zero the result is obtained only

on the even by αthe bump functions ψ0(z−α).

Therefore, we have the solution of Cauchy prob-

lem for Shrödinger equation on small time interval ∆t

ψ∆t(z) = ψ0(z)−

−i∆t

π −ε

−∞

∞

ε!α−2ψ0(z−α)dαε→0

Note, that «deinterlaced» photon, according to this

formula, being in any localized state ψ0(z)at t= 0,

at the very first moment it goes beyond the limits of

the classic light cone z=t,t < ∆t, on which it was

located at t= 0, and filling at once whole coordinate

space outside the small ε-vicinity of the origin of co-

ordinates.

Therefore the Hamiltonian of such functional is

generalized function

Hε(z) = 1

π









z−2,−∞ < z < −ε

0,−ε≤z≤ε

z−2, ε < z < ∞

Remark, that the constructed solution ψ∆t(z)conve-

niently to take as the «deinterlaced» photon state, in

which mirrors are introduced, that are mentioned in

«Introduction», at points ±ε, since such mirrors does

not deform such quantum field.

Thus the wave functional, in which mirrors are in-

troduced,

∆tψ0(z) =

=−i∆t

π −ε

−∞

∞

ε!α−2ψ0(z−α)dαε=0

=ψε(z)|ε=0

(6)

and 2εis distance between mirrors.

Moreover, since the operator ∂

∂ε (meaning the si-

multaneous moving asunder the mirrors), acting on

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DOI: 10.37394/232020.2023.3.9

Beilinson A. A.

E-ISSN: 2732-9941

Volume 3, 2023

wave functional ψε(z)

∂

∂ε

−i∆t

π −ε

−∞

∞

ε!α−2ψ0(z−α)dα =

=i∆t

π(ε

←−2+ε

→−2)α−2ψ0(z−α)

(7)

(arrows mean direction of forces acting on left and

right mirror) has the meaning of the force operator

acting on mirrors from the side of the quantum field

(Casimir forces) and moving asunder the mirrors.

Here the a negative energy of the quantum field of

«deinterlaced» photon corresponds to Casimir forces,

contrary to positive energy of the quantum field of

usual photon.

Remark, that the operator ∂

∂ε acts on an element

of the Hilbert space ψε(z)(see (6)), is interpreted as

a projection operator of this state to one of summand

(mutually non-orthogonal vectors of Hilbert space) of

integral sum.

Therefore the average of Casimir operator ∂

∂ε on

quantum field in the state ψε(z)is

∂

∂ε =R¯

ψε(z)∂

∂ε ψε(z)dz

R¯

ψε(z)ψε(z)dz =

=R¯

ψε(z)(ε

←−2+ε

→−2)ψ0(z−ε)dz

R¯

ψε(z)ψε(z)dz .

For calculation of this fraction we will write the inte-

grand vectors in the trigonometric (orthonormal) ba-

sis. Since

R¯

ψ0(z−α)ψ0(z−α′)dz =

=Rexp−ip(α−α′)˜

ψ0(p)

2dp,

we have (see (7))

∂

∂ε = (ε

←−2+ε

→−2)×

× R −ε

−∞

∞

ε!α−2e−ip(α−ε)˜

ψ0(p)

2dα dp!×

× R −ε

−∞

∞

ε!α−2exp(−ipα)˜

ψ0(p)

2×

× −ε

−∞

∞

ε!α′−2exp(−ipα′)dα dα′dp!−1

= (ε

←−2+ε

→−2)×

×R −ε

−∞

∞

ε!α−2exp(−ip(α−ε))|˜

ψ0(p)|2dα dp

R −ε

−∞

∞

ε!α−2exp(−ipα)dα

|˜

ψ0(p)|2dp

= (ε

←−2+ε

→−2)×

∞

α−2cos(pα)cos(pε)|˜

ψ0(p)|2dα dp

∞

0∞

α−2cos(pα)dα2

|˜

ψ0(p)|2dp

Since this fraction is interpreted in the geometry of

a Hilbert space as the square cosine of the angle be-

tween the vectors of this space ∂

∂ε ψε(z)and ψε(z), the

value of this fraction does not depend on the choice of

bump function ψ0(z), that is why any bump function

can be used to numerically find the value of this frac-

tion, for example, ψ0(z) = exp(−z2).

Recall that the real quantum e.-m. field in the

state of «deinterlaced» photons (taking into account

the states of photon-antiphoton) contains two inde-

pendent components.

Recall also that the influence of the field corre-

sponding to element dt of a «Feynman path» z(t)

is taken into account, while the whole plane (z=

0) is exist, which is why we have that «Casimir

forces»corresponding each area unit of the plane z=

0is

2∂

∂ε !−2

At the same time, there is a continuous «tissue» of the

quantum field ψ0(z)of «deinterlaced» photon, where

the Casimir forces do not appears absolutely. That is

interpreted as the presence of negative pressure forces

into this environment stretching that «tissue». There-

fore the Casimir forces measured in known experi-

ment are exactly equal and opposite to the considered

ones in present work.

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Beilinson A. A.

E-ISSN: 2732-9941

Volume 3, 2023

5 Conclusion

The work shows that the real existence of Casimir

forces indicates the reality of the existence of two

unitary equivalent, but fundamentally different phys-

ically forms of the quantum e.-m. field, one of which

(the known) is described by the Maxwell–Majorana

equations with the solutions inside the light cone, con-

trary to other form — the quantum field, the field of

«deinterlaced» photon, existing outside the light cone

and responsible for Casimir forces.

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Contribution of individual authors to

the creation of a scientific article

(ghostwriting policy)

The authors equally contributed in the present re-

search, at all stages from the formulation of the prob-

lem to the final findings and solution.

Sources of funding for research

presented in a scientific article or

scientific article itself

No funding was received for conducting this study.

Conflict of Interest

The authors have no conflicts of interest to declare

that are relevant to the content of this article.

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PROOF

DOI: 10.37394/232020.2023.3.9

Beilinson A. A.

E-ISSN: 2732-9941

Volume 3, 2023