## 1. Introduction

Linearized single photon Hamiltonian is used for the analysis of its features in coordinate systems of various geometries. As it could have been expected, based on the general theory of relativity, it turned out that space geometry and physical features are closely interrelated. In Cartesian’s coordinates single photons are spatial plane waves, while in cylindrical coordinates they are one-dimensional plane waves the amplitudes of which falls in planes normal to the direction of propagation. The most general information on single photon characteristics has been obtained by the analysis in spherical coordinates. The analysis in this system has shown that single photon spin essentially influences its behavior and that the wave functions of single photon can be normalized for zero orbital momentum, only.

A free photon Hamiltonian is linearized in the second part of this paper using Pauli’s matrices. Based on the correspondence of Pauli’s matrices kinematics and the kinematics of spin operators, it has been proved that a free photon integral of motion is a sum of orbital momentum and spin momentum for a half one spin. Linearized Hamiltonian represents a bilinear form of products of spin and momentum operators. Unitary transformation of this form results in an equivalent Hamiltonian, which has been analyzed by the method of Green’s functions. The evaluated Green’s function has given possibility for interpretation of photon reflection as a transformation of photon to anti-photon with energy change equal to double energy of photon and for spin change equal to Dirac’s constant. Since photon is relativistic quantum object the exact determining of its characteristics is impossible. It is the reason for series of experimental works in which photon orbital momentum, which is not integral of motion, was investigated. The exposed theory was compared to the mentioned experiments and in some elements the satisfactory agreement was found.

## 2. Eigen-problem of single photon Hamiltonian

In the first part of this work the eigen-problem of single photon Hamiltonian was formulated and solutions were proposed. Based on the general theory of relativity, it turned out that space geometry and physical features are closely interrelated. Because of that the analyses was provided in Cartesian’s, cylindrical and spherical coordinate systems.

### 2.1. Introduction

Classical expression for free photon energy is:

where * c* is the light velocity in vacuum and

p

_{x},

p

_{y}and

p

_{z}are the components of photon momentum. If instead of classical momentum components we use quantum-mechanical operators

This Hamiltonian is not a linear operator that contradicts the principle of superposition (Gottifried, 2003; Kadin, 2005). Klein and Gordon (Sapaznjikov, 1983) skirted this problem solving the eigen-problem of square of Hamiltonian (2):

since the square of Hamiltonian is a linear operator. This approach has given satisfactory description of single photon behaving. Up to now it is considered that this approach gives real picture of photon. Here will be demonstrated that Kline–Gordon picture of photon is incomplete.

Here we shall try to examine single photon behavior by means of linearized Hamiltonian (2). Linearization procedure is analogous to the procedure that was used by Dirac’s in the analysis of relativistic electron Hamiltonian (Dirac, 1958). We shall take that

i.e. we shall transform the sum of squares into the square of the sum using

It is easy to show (Tošić, et al., 2008; Delić, et al., 2008) that (5) conditions are fulfilled by Pauli’s matrices

Combining (6), (4) and (2), we obtain linearized photon Hamiltonian which completely reproduces the quantum nature of light (Holbrow, et al., 2001; Torn, et al., 2004) in the form:

^{.}

Since linearized Hamiltonian is a 2×2 matrix, photon eigen-states must be columns and rows which two components. Operators of other physical quantities must be represented in the form of diagonal 2×2 matrices.

At the end of this presentation, it is important to underline the orbital momentum operator

In further the eigen-problem of linearized single photon Hamiltonian will be analyzed in Cartesian’s, cylindrical and spherical coordinates.

### 2.2. Photons in Cartesian's picture

The eigen-problem of single photon Hamiltonian in Cartesian coordinates (we shall take it with plus sign) has the following form:

wherefrom we obtain the following system of equations from:

where

Since the operators

In the same manner, from (9) and (11), we come to the relation:

The two last relations are of identical form and can be substituted by one unique relation:

If we take in (14) that

which is fulfilled if:

Equations (16) can be easily solved and each of them has two linearly independent particular integrals:

Based on these expressions, we conclude that eigen-vector of single photon

Since * δ*–function, wherefrom follows:

Solving these integrals, we come to: 2 ^{2} (2* π*)

^{3}= 1, wherefrom we get the normalized single photon eigen-vector as:

As it can be seen from (20), the components of single photon eigen-vector are progressive plane wave ~

### 2.3. Photons in cylindrical picture

In this section of first part of the paper we are going to analyze the same problem in cylindrical coordinates. Since solving of partial equation of

In order to examine this problem, we shall start from the equation (14) in which Laplacian * ρ*,

*,*φ

*) where*z

and therefore (14) with

The square of wave vector * k* will be separated into two parts

By the substitution:

the equation (22) reduces to:

This equation is fulfilled if:

Now we separate the variables by substitution:

after which, the (25) goes over to:

Introduction of the variables separation constant ^{2} represents generalization with respect to approach used in previous section. Since the function * m* is integer, i.e.

Relation (27) is separated into two differential equations:

The equation (25) has two particular integrals:

while the solution of the equation (28) is:

By the substitution of argument

and taking that* m*:

It means that the solution of (28) is the * m*–order Bessel’s function:

J

_{m}, i.e.

Taking into account (29), (30) and (33), we obtain the components of single photon eigen-vector:

Since * q* and

k

_{z}are continuous variables, while

*is a discrete one the normalization of eigen-vector must be done partially to*m

*–functions and partially to Kronecker’s symbol. It means that normalization condition is the following:*δ

Using formula for normalization of Bessel functions with integer index (Korn & Korn, 1961):

the normalization condition reduces into:

The first component Φ_{1} corresponds to photon (velocity +* c*), while second component Φ

_{2}corresponds to anti-photon (velocity –

*). From this formula we conclude that single photon eigen-vector components are progressive and regressive plane waves along*c

*-axis. In the (*z

*,*x

*) planes components change periodically with polar angle*y

*and decrease by the rule*φ

ρ

^{-1/2}with distance between the axis and envelope of cylinder. The last is concluded on the basis of asymptotic behaving of Bessel’s functions (Korn & Korn, 1961):

### 2.4. Photon in spherical picture

The analysis of single photon eigen-problem in spherical coordinates, as it well be shown later, requires introduction of two variable separation parameters. We start from the equation (14), where the Laplace’s operator will be written down in spherical coordinates (* r*,

*,*θ

*), where*φ

It means that (14), with

In the first stage of variables separation, we shall take that:

after which substitution into (37), it goes over to:

where Λ^{2} is the variable separation parameter. Double equality in (39) gives two equations:

It should be noted that equation (40) represents eigen-problem of ^{2} determines orbital quantum numbers. In this equation we shall take that:

after this substitution, which goes over to:

In this double equality the variable separation parameter * m* must be integer since the solution

When the solution (43) is:

the equation (43) is associated Legendre’s equation (Gottifried, 2003; Davidov, 1963). The complete procedure of solving of this equation cannot be found in literature. Instead of the general solving procedure of the equation (43) is solved for * m* = 0. Its solutions are Legendre’s polynomials (Korn & Korn, 1961; Janke, et al., 1960). Differentiating these polynomials

*-th times it was possible to conclude that solution (43) can be expressed through*m

*-th Legendre’s polynomials derivations.*m

In order to avoid such an artificial solving of the equation (43), we shall expose, briefly, its solving by means of potential series. This solving procedure may be comprehended as methodological contribution of this part of the paper. At the first stage, we translate the equation (43) into algebraic form by means of substitution of argument

The term * U* is an arbitrary function, the equation (45) reduces to the same form but with arbitrary constant in linear function with is multiplied by first derivative of

*function. This arbitrary coefficient will be taken in the form*V

*is arbitrary. Arbitrary constant*s

*will be determined in a way which eliminates the term*s

*function. By the described strategy the (45) becomes:*V

This equation is suitable for solving by means of potential series. Arbitrary function * U* is given by

*has the form:*T

Since * s* must not be negative since

*would then have singularities in*T

The solution of this equation will looked for in the form of potential series:

after which substitution in (48) we obtain recurrent formula for series coefficients:

Here arises a dilemma whether to leave the whole series or to cut it and retain a polynomial instead of series. In order to solve this dilemma, we shall analyze a special case of formula (50) when

wherefrom it turns out that

From this formula is obvious that the series has singularities for * ζ* = ±1. This resolves above mentioned dilemma: the series must be cut and the polynomial obtained in this way must be taken as solution. From the formula (50) it is clear that the series will be cut if:

Now is clear that the series is cut when* m* per module must not exceed

*:*l

*function is expressed as:*T

The product of functions (44) and (54) normalized per angles gives spherical harmonics (Gottifried, 2003; Davidov, 1963):

Finally we shall solve the equation (40) in which Λ^{2} is substituted by

Substituting the function * R* with

*by*r

*/*ρ

*, we translate last equation into Bessel’s equation (Korn & Korn, 1961; Janke, et al., 1960) with*k

It is necessary for further to quote behaving of Bessel’s functions with half integer indices. It can be easily shown that:

As well as using recurrent formula for Bessel’s functions (Janke, et al., 1960):

and taking that

Due to the factor * A* which are proportional to

The very important conclusion of this analysis is: only free photons with zero orbital momentum have chances to be normalized exist. For

We shall now examine whether the components of photon eigen-vector proportional to _{1} and _{2} can be normalized. Those components are:

The normalization condition is the following:

It is not difficult to show that:* l* = 0 photon eigen-vector cannot be normalized.

The last possibility for normalization free photons eigen-vector is so called box quantization method. In this method the sphere is substituted by cube enveloping it and cyclic boundary conditions are required:

Since

It is seen that the first harmonic of electromagnetic waves has the wave length equal to the cube edge.

Photon energy is determined in the standard way:

This expression for energy is in full accordance with Plank’s hypothesis (Planck, 1901).

In the normalization condition (63) the following translations has to be used:

Combining this and (63) we obtain that the normalization constant is

As it can be seen the analysis of single photon eigen-problem in spherical coordinates has shown it orbital momentum of photon is equal to zero and that the spin

## 3. Free photon as a system with complex internal dynamics

In the second part of this work the free photon Hamiltonian will be linearized using Pauli’s matrices. Based on the correspondence of Pauli matrices kinematics and the kinematics of spin operators, the unitary transformation of this form (equivalent Hamiltonian), will be analyzed by the method of Green’s functions. Since photon is relativistic quantum object the exact determining of its characteristics is impossible. It is the reason for series of experimental works in which photon orbital momentum, which is not integral of motion, will be theoretically investigated.

### 3.1. Introduction

The fact that photon Hamiltonian is not a linear operator has a set of consequences that have not been studied sufficiently so far. The main reason is that photon characteristics have been mainly examined by means of Klein-Gordon’s equation (Gottifried, 2003; Davidov, 1963; Messiah, 1970; Davydov, 1976), which represents eigen-problem of photon Hamiltonian square. In this part of our paper we shall linearized photon Hamiltonian and examine some of photon characteristics witch follow from linearized Hamiltonian. The analogy with Dirac’s approach to the problem of electrons will be used (Gottifried, 2003; Dirac, 1958). Firstly will be examined integrals of motion of free photon and will be shown that the photon integral of motion is not orbital momentum. It will be shown that the integral of motion is total momentum being the sun of orbital one and spin momentum.

The evaluated Green’s function has given possibility for interpretation of photon reflection as a transformation of photon to anti-photon with energy change equal to double energy of photon and for spin change equal to Dirac’s constant (Dirac, 1958; Messiah, 1970). Since photon is relativistic quantum object the exact determining of its characteristics is impossible.

The discussion of obtained results and their comparison to the experimental data will be done at the last part.

### 3.2. Linearized photon Hamiltonian

We shall not deal with this eigen-problem in further of this paper. Instead of this we shall look for integrals of motion, i.e. those operators that commute with free-photon Hamiltonian (7). It is obvious that any function depending on momentum components represents an integral of motion, but this fact is not of physical interest.

It is of particular importance whether orbital momentum:

is photon integral of motion, since in non-relativistic quantum mechanics operator

If we use commutation relations for components of radius vector and the components of momentum:

based on which it follows that orbital momentum is not a free photon integral of motion.

It should be pointed out that signs in (70) are obtained on the basis of obvious symmetry properties

In order to find some rotation characteristics that commute with a free photon Hamiltonian, we shall first show that commutation relations for matrices

while commutation relations for spin components (Dirac, 1958; Messiah, 1970):

are very similar to (71). Comparing (71) to (72) we can establish the correspondence between spin operator components and matrices

Commutators of matrices

We shall now look for a commutator of component

For lower signs in formulas (70) and (74)

this corresponds to negative photon energies, i.e. corresponds to

, we have:It can be proved, in the same manner, that both * y* and

*components of total momentum*z

*and*y

*components of total momentum*z

The final conclusion is the following: total momentum

If spin is

For spin

(in the last stage of the upper proof the relations (72) from section 2.1 were used). Consequently, we can conclude that free photon integral of motion represents a total momentum which is the sum of orbital momentum and spin momentum which corresponds to the case when

In the same way can be concluded that anti-photon integral of motion is the sum of orbital momentum and spin momentum which corresponds to spin* ,* where

In nonrelativistic quantum mechanics (Gottifried, 2003; Davidov, 1963) the conclusion that

Considering the correspondence (73), photon Hamiltonian which is given by

The obtained form of photon Hamiltonian, which includes operators of translation moment

### 3.3. Unitary transformation of photon Hamiltonian

Photon Hamiltonian (78) represents bilinear form in which photon momentum operators are multiplied by spin operators. Since momentum characterizes translation photon motion, and spin characterizes rotation, it is obvious that the internal dynamic structure of a photon is determined by both its translation and rotation characteristics, and that their interaction – considering the form of Hamiltonian (78), leads to hybridization of excitations (Agranovich, 2009). Spin operators in (78) correspond to spin * S* = 1/2 and its can then is represented by Pauli’s operators in the following manner (Tyablikov, 1967):

Pauli’s operators fulfill commutation relations:

After substitution of (79) in (78) (in this formula sign + is retained), we obtain the following form of Hamiltonian:

This conversion to Pauli operators has been made because the physical picture of processes is clearer through operator’s creation and annihilation of excitation.

Operators of moments are linear in operators of creation and annihilation of photon:

where:

and * ρ* and

*are real parameters.*λ

Equivalent Hamiltonian is found using Weil’s identity (Tošić, 1978):

It has included the terms of the following type:* λ* has been determined so that the member proportional to

*–*P

P

^{+}disappear from equivalent Hamiltonian. The final result of the described procedure is as follows:

where

where are:

We shall further analyze free photon behavior using method of Green’s functions (Tyablikov, 1967; Tošić, 1978; Rickayzen, 1980; Mahan, 1990; Šetrajčić, et al., 2008). Hamiltonian _{0} is irrelevant in Green function techniques. Starting Hamiltonian

### 3.4. Green’s function of free photons

Since Pauli operators figure in

where

Differentiating * P*, we come to the following equation:

The Green’s function of type:

Using the same procedure, for defining function

where:

with defining following equation:

In differential equations (88), (90) and (92), Furrier’s transformations time-frequency are then made:

so we obtain the system of algebraic equations:

Solving this system of equations, we find that:

where:

In order to determine spectral intensity of function Γ, it is necessary to break down the right side of the formula (87) into common fractions. So, we obtain the following:

where:

and using Dirac’s formula:

where P.V. denotes principal value of integral, we find the explicit expression for spectral intensity:

Now we can defined the expression for correlation function of a free photon as:

Next, we can calculate expression for concentration of spin excitations of a free photon. It is obtained from (101), if we take in it that * t* = 0, i.e.

Combining formulae for * a* over formula (86), and

E

_{0}from (96), and converting to sphere coordinate system, we find that:

^{xxxx}

^{xxxx}

In accordance with this and formula (102), we get the following expression for ordering parameter of spin subsystem in a free photon:

The set of results of this section requires some explanations. The most interesting results is that energy for spin translation from * ħk*. So we obtain the change of photon momentum

^{’}s function

*and this is eigen-value of spin*ħ

*= 1. This is the reason for behaving of photon as particle with spin*s

*= 1.*s

The polar and azimuthally dependences of ordering parameter comes from the fact that incident bean must not be always orthogonal to the plane of measuring device.

## 4. Conclusions

The analysis of single photon behaving in coordinate systems of various geometries has shown the following:

The last result shows that linearization of photon Hamiltonian gives more complete picture of single photon than Kline-Gordon’s approach.

Concluding the exposed analysis we shall try to connect the results obtained in series of experimental investigation of photon orbital momentum (Beth, 1936; Leach, et al., 2002; Allen, et al., 1992; Allen, 1966; He, et al., 1995; Friese, et al., 1996; Markoski, et al., 2008; van Enk & Nienhuis, 2007; Santamato, et al., 1988; O’Neil, et al., 2002; Volke-Sepulveda, et al., 2002). We shall not describe all quoted experiments. Instead of it we shall describe the essential idea: the orbital momentum of photon was determines from the changes of torque of rotating particles. These changes where lied in some interval, so that the values of orbital momentum have had determined dispersion. As it vas said at the end of first section, such result is expectable for relativistic objects, in this case for photons. The azimuthally dependence of measured results is also predicted by the theory exposed in last Section.

Ending this analysis it should by noticed out that on the bases of given analysis the photon reflection can be considered as a transformation of photons to anti-photons.

## Acknowledgments

Investigations whose results are presented in this paper were partially supported by the Serbian Ministry of Sciences (Grant No 141044A) and by the Ministry of Sciences of Republic of Srpska.

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## Notes

- this corresponds to negative photon energies, i.e. corresponds to