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The Dirac equation in Rindler space: A pedagogical introduction

A pedagogical introduction to the Dirac equation for massive particles in Rindler space is presented. The spin connection coefficients are explicitly derived using techniques from general relativity. We then apply the Lagrange-Green identity to greatly simplify calculation of the inner products needed to normalize the states. Finally, the Bogolubov coefficients relating the Rindler and Minkowski modes are derived in an intuitive manner. These derivations are useful for students interested in learning about quantum field theory in a curved space-time.


Introduction
One of the most dramatic results in the early attempts to marry quantum field theory to general relativity was the discovery by Hawking that black holes radiate 1 . This interesting result was soon followed by the discovery by Unruh 2 that a uniformly accelerated detector would be immersed in a bath of thermal radiation with temperature , where a is the acceleration of the observer ( 1 kc ==). Moreover, an accelerated observer in flat space-time has his or her own event horizon. The result of Unruh can be encapsulated by saying each observer has his or her own vacuum.
These celebrated results, together with a bit of simplicity that flat space-time brings, have made Rindler space a very interesting arena within which to study quantum field theory. While most treatments have focused on the scalar field, one effort many years ago briefly showed that a uniformly accelerated observer would see a flux of massive Dirac particles 3 .
Due to the recent interest in quantum information theory in a relativistic context, in particular the consideration of teleportation with an accelerated observer 4 , in this primarily pedagogical paper we recall the main results associated with Dirac particles in Rindler space. Our motivation is two fold. First, this paper is designed to assist students interested in studying quantum field theory in a curved space-time. In addition, we aim to put the necessary machinery on the table so that a later work can consider quantum information theory with accelerated observers using a complete relativistic framework.
The outline of this paper is as follows. After briefly describing Rindler coordinates and the form of the Dirac equation in a curved space-time, we derive the spin connection coefficients by considering the geodesic equation. In the next section, we address the crucial problem of normalizing the states, which are given in terms of Hankel functions. This is a non-trivial problem. This paper addresses that fact by applying the Lagrange-Green identity to greatly simplify the calculation. To our knowledge, this approach has not been used in this case. Finally, after normalizing the states, we show how to write the Minkowski wave functions in terms of the Rindler wave functions and vice versa. This allows us to conclude the paper by deriving the Bogolubov coefficients.

Rindler Coordinates
In this paper we consider the familiar case of a uniformly accelerated observer in Minkowski space-time. As shown in Fig. 1, we consider two-dimensional Minkowski space with coordinates ( ) , tz divided into four regions. These are the right Rindler wedge (region I), the left Rindler wedge (region II) and the future (F) and past (P Since the calculations described in this paper will be similar for all four regions, we focus on region I, which we denote as Rindler space. In this region, starting with the two-dimensional Minkowski line element 222 dsdtdz =− and using (1), it is easy to show that the line element takes the form Note that v is a timelike coordinate and u is a spacelike coordinate-in region I. This will change in the other regions. Using this line element we can read off the components of the metric tensor.
These can be arranged in a matrix as follows 0 The components of g µν can be found easily by inverting (3). We find that 1/0 01 u g µν  =  −  (4) In the next section, we proceed to derive the form of the Dirac equation in Rindler space.

The Dirac equation in Rindler space
Given general curvilinear coordinates with metric g µν , the Dirac equation is written as 5 where the µ Γ are known as the spin connection coefficients.
Using g µν to raise indices it can be shown that Our next task on the road to computing the spin connection coefficients is to turn to the problem of fin ding the form of the Christoffel symbols. To do this we begin with a brief study of the geodesic equation. General relativity tells us that the paths of particles moving without the influence of any external forces are determined by the following equation where the derivatives indicated by a dot are taken with respect to an affine parameter. The explicit form of the Christoffel symbols and therefore of the spin connection coefficients can be determined in a fairly painless fashion by applying a Lagrangian procedure 6 . We start by defining a quantity which we label as K Our next task is to write down the solutions to (24) and normalize the states.

Normalization with the Lagrange-Green Identity
The ultimate goal of the present exercise is to relate the complete Minkowski wave function to the Rindler wave functions in all four regions. In order to do this properly we will need to normalize the states. This turns out to be fairly mathematically difficult and may even appear a bit mysterious to most readers. However, as we'll see below, this process is greatly simplified by applying the Lagrange-Green identity.
The first step is to look at (24) by thinking of the Dirac equation as a differential operator. We will call that operator L and note by looking at the right hand side of the Dirac equation (24), we can write L as Of course we can view the Dirac equation as an operator and look at the left side of (24) as well.
In that case we can write Stationary states will be of the form ( ) iv eu ω ψψ − = , in which case the application of (27) immediately yields To proceed further we will need the explicit form of the states. In this paper we consider the spinup case whereψ is a four component column vector given by 3 The ( ) mu ω φ ± satisfy the following differential equation We can calculate this integral by applying the Lagrange-Green identity (described in Appendix B). We begin by examining the adjoint of the differential operator given in (26). In particular, we wish to show that This relation corresponds to that described by (90) in the appendix. In due course we will see the value of this relation by noting that when considering inner products of states, the presence of the will allow us to write down the results of the relevant integrals almost by inspection.
Let's consider the expression ( ) † Lψφ in detail. Using (24) and (26), we see that this is We approach the goal of rewriting this expression so that L is applied to φ by considering each piece on the right hand side of (35) in turn. Beginning with the first term, we have However, notice that And so we can write ( ) Putting these results together gives However, looking back at the definition of L, we have Therefore we have shown that Notice that using the result of the Lagrange-Green identity as applied to the Dirac equation, we have demonstrated that We'll see very shortly that it is now possible to compute the normalization integral given in (33) using (44).

Now we set
The presence of the term 2 z π ensures that as z →∞, these terms will vanish. Therefore (47) In Appendix A we will show that The minus sign in (50) cancels the one here. Moreover, we obtain the exact same result modulo the minus sign for the other term in (48). Adding these results together we find that the norma lization is given by

Expressing Minkowski wave functions in terms of Rindler wavefunctions
The Minkowski wave function is defined over the entire ( ) , tz plane. Therefore in order to express it in terms of Rindler wave functions, we need to know the form of the Rindler states in all four regions. We now describe the states in regions II, F, and P and then piece them together to write down the Minkowski wave function.
In order to distinguish the states of regions I, II, F, and P we will adopt the notation ,,, and IIIFP ωωωω ψψψψ respectively. Furthermore, it is necessary to distinguish between positive and negative frequency modes in the future and past regions. These can be denoted by where ( ) ± denotes particle/anti-particle mode respectively.
The derivation and normalization of the Rindler states in the other regions is similar to that used in the previous section. So we simply state the results, continuing to examine the spin-up case. In the future region, the Dirac equation assumes a different form because u becomes a timelike coordinate and v becomes a spacelike coordinate. The spin-up state in the future region is given by 3 As an aside, note that strictly speaking it is necessary to write down the wave functions so that they are only defined in the appropriate region. This can be done using the Heaviside step function. For our purposes, it won't be necessary to worry about this explicitly. We simply note that each Rindler wave function is defined in its given region and is zero elsewhere. Now let's turn to the problem of expressing the Minkowski states in terms of the Rindler states. In a previous work 3 a complex argument involving source terms on the light cone (which is bound to confuse the student) is used to suggest the form of the Minkowski wave function. We instead take a more heuristic approach based on simple mathematical arguments.
Observing that the Minkowski modes must cover the entire ( ) Now, for the (+) frequency mode, we have 1 i e π −= . Therefore we can proceed as follows

Calculating the Bogolubov coefficients
In this section we will determine the Bogolubov coefficients which allow us to write the Rindler wave function in terms of positive and negative frequency Minkowski modes. These were originally stated by Soffel 3 , and can be obtained by simple algebra. The key concept in this derivation is that the Rindler observer in region I is causally disconnected from region II and vice versa. If we focus on an observer in region I, this means that we seek a combination of ω ψ + and ω ψ − that reflects this fact by eliminating II ω ψ .
Let's denote the wave function as seen by this Rindler observer in region I by RI ω ψ . The linear combination of ω ψ + and ω ψ − we seek is ( ) It is necessary to normalize this state. Using (78) we find that A similar procedure can be applied to the state in the left wedge, by noting that the observer in region II is causally disconnected from region I. This is a good exercise for the student.

Conclusion
This paper has provided a pedagogical introduction to working with massive Dirac particles in Rindler space. In addition, we have applied the Lagrange-Green identity which greatly simplifies the problem of normalizing the states, obtaining the exact result found in the literature. To the knowledge of the author this approach has not be used before. We concluded by applying simple mathematical arguments to express the Minkowski wave functions in terms of the Rindler states.
The "machinery" described in this paper can be applied to further research. In particular, it can be used to give a full relativistic treatment to problems in quantum information theory that involve massive Dirac particles and an accelerated observer. We intend to pursue this line of research in a future paper. The value of doing so has been demonstrated by recent interest expressed in the properties of entangled particles in a gravitational field 9 .

Appendix A: Mathematical Preliminaries
We now prove three crucial propositions that were used to calculate the normalization of the Rindler states. In the following, we will consider functions ( ) x φ on the real numbers that are infinitely differentiable and that vanish outside of some region. These functions are called test functions are denoted by writing ( ) We begin by considering the well-known Riemann-Lebesgue lemma, which basically states that the Fourier coefficients ˆk f of a periodic integrable function vanish as k →∞.