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We study the quantum theory of the mass-less vector fields on the Rindler space. We evaluate the Bogoliubov coefficients by means of a new technique based upon the use of light-front coordinates and Mellin transform. We briefly comment about the ensuing Unruh effect and its consequences.

The production, propagation and detection of real photons in a non-inertial reference frame are a highly nontrivial subject [

It is the aim of the present short note to fill this lack and to provide a fully consistent Lorentz and gauge invariant quantum theory for the lineal, i.e. on a one dimensional spatial real line, radiation field. In so doing, thanks to a new method based upon the use of the Mellin transforms and the light-front system of coordinates, which has been recently developed by Aref’eva and Volovich [

^{1}Throughout this note we shall use natural units

In this Section we aim to briefly collect and recall the most useful systems of coordinates, which are utmost suitable to describe the field dynamics for a uniformly accelerated Observer in the Rindler space [

and metric

where

are the standard light-front coordinates. If we perform the non-inertial coordinate transformation in the restricted space-like region

with

then the line element of Equation (1) becomes

Moreover we can readily obtain the inversion formulæ for

whence we get

in such a manner that if we set

Finally we can readily find the transformation formulæfor the differential operators, viz.

The curved coordinates

Since

the asymptotic of which are the null rays

in such a manner that hyperbolæ of large negative ξ, i.e. close to the Rindler horizon

As a further quite useful example of curved coordinate system to label the two dimensional Rindler space, consider once again the sub-spaces R and L of the Minkowskian space

where the plus and minus signs refer to the right and left Rindler’s wedges respectively, while the line element takes the forms

whence we obtain the metric tensors

so that

Moreover, after setting

we readily get the transformation matrix

where

and specifically, for the present case of a uniformly accelerated Observer,

Hence we can eventually write the following chain equality

For any contravariant vector field in the Minkowski space we can derive the corresponding vector field in the Rindler space, viz.,

Consider for example the complete orthogonal set of the mass-less vector normal modes in the 2-dimensional Minkowski space

where the dual pair of constant light-like polarization vectors is provided by

while the scalar wave functions read

The Minkowskian space vector normal modes do fulfill

where the invariant inner product is defined by

The complete and orthogonal set of the mass-less vector normal modes in the right Rindler wedge R can be readily obtained from Equations (21) and (25). To this concern it is expedient to introduce the accelerated polarization vectors

in such a manner that we eventually come to

where

with

The manifestly covariant quantization of the free radiation field on a two dimensional Minkowskian space

where

It turns out that the following general canonical commutation relations hold true: namely,

where the mass-less Pauli-Jordan real and odd distribution is provided by

whereas

an integral-differential representation being given by

An explicit expression can be obtained as follows. Let us first consider the positive and negative Wightman functions, i.e. the positive and negative parts of the Pauli-Jordan distribution, in the two dimensional Minkowskian space

whence it is clear that the positive and negative parts of the Pauli-Jordan commutator are complex conjugate quantities

Consider now the integral for

and change the variable

Here

in such a way that we can write

Now we can use the integral representations of the Bessel functions of real and imaginary arguments [

and thereby

so that

while the mass-less dipole-ghost in two space-time dimensions becomes

It is a simple and instructive exercise to verify the compatibility between the above general covariant canonical commutation relations and the equations of motion, for

together with

Moreover one can readily check that the initial conditions fulfilled by the above general covariant canonical commutation relations are

all the remaining equal-time commutation relations being equal to zero. The most general solution of the canonical commutation relations in the non-homogeneous Lorenz gauge can be written in the form

where

all the other commutators vanishing. The canonical commutation relations indicates that the Fock space is of indefinite metric, so that a physical Hilbert sub-space with semi-definite metric is selected by the subsidiary condition

where

is the positive frequency part of the auxiliary scalar field operator. It follows therefrom that, for instance, the 1-particle states

Things greatly and neatly simplify in the Feynman gauge

we can recast the covariant wave equations and the non-homogeneous Lorenz condition in the light-front form

It turns out that the mass-less vector wave equations and gauge condition in a two dimensional Minkowskian space possess the set of light-like solutions

where

does represent the standard orthogonal set of mass-less scalar modes. As we have

it follows that the Lorenz condition is satisfied only for the longitudinal normal modes, viz.,

Notice that the polarization vector

the other light-like bi-vector

In order to set up a complete orthogonal basis of mass-less vector modes on the Minkowskian space

one has to introduce the dual light-like polarization vector

It turns out that all the normal modes (61) are of positive frequencies with respect to the time-like Killing vector

It is also useful to introduce the light-front components of the light-like polarization vectors that read

The normal modes with

It turns out that in the standard instant-form coordinates

Conversely, for

which shows that the trigonometric functions of the temporal light-front variable

It is worthwhile to remark that the mass-less normal modes

while the polarized vector normal modes

which fulfill

The vector normal modes

Putting altogether we can write the light-front components of the Feynman gauge vector potential in the form

together with

Moreover, from Equation (53) we get the normal mode expansion of the auxiliary scalar field

It is worthwhile to remark that the above expressions manifestly satisfy both the D’Alembert wave equation as well as the non-homogeneous Lorenz condition because

while the electric field strength is provided by

It is very convenient to write

in such a manner that one gets a representation for the solutions of the wave equations and the Feynman gauge subsidiary condition in the Rindler’s regions R and L, the two regions being interchanged by parity and time reversal symmetry. Actually, in the Rindler’s regions L and R one may adopt an alternative expansion based upon the Rindler’s counterparts

for which there exist normal mode solutions

It follows that from Equation (3) we eventually find for

where

The relation between the normal modes expansions of the quantum fields in the Minkowski and Rindler spaces is well known [

which admits analytic continuation to a meromorphic function in the whole complex plane with simple poles at

where

Moreover we get

where use has been made of the Minkowski set (76). From the relation

we definitely obtain

that eventually yields the relationships between the holomorphic amplitudes of the Lorenz gauge potential on the Minkowskian space

in full accordance with [

that yields

which actually corresponds to Bose-Einstein thermal distributions at the equilibrium temperature

where use has been made of the Minkowski set (73). Hence we definitely obtain

and thereby

In this short note we have presented the quantum theory of the lineal, i.e. one dimensional in space, radiation field in a uniformly accelerated reference frame referred to Rindler’s curved coordinates. The pair of physical and nonphysical radiation fields, which must be introduced in a diffeomorphism and gauge invariant quantum theory on a flat space-time, appear to be described by null norm quantum fields, in such a manner that a subsidiary condition must be introduced to select the physical Hilbert sub-space, the quantum states of which are of positive semi-definite norm according to [

The work has been supported by the Department of Physics and Astronomy of the University of Bologna (DIFA).

RobertoSoldati,11,CaterinaSpecchia, (2015) On the Massless Vector Fields in a Rindler Space. Journal of Modern Physics,06,1743-1755. doi: 10.4236/jmp.2015.612176