Transforming quantum states between reference frames

In the 1970s, Fulling, Davis, and Unruh have shown that a quantum mechanical state must be described differently in different reference frames; otherwise, quantum mechanics would contain contradictions. We present a simple method for transforming any quantum state between the Minkowski and Rindler reference frames. We show that a Wigner-like distribution, commonly used in quantum optics, is useful for treating this problem. To illustrate our method, we transform the Minkowski vacuum and number states into Rindler space, and transform the Rindler vacuum into Minkowski space, as examples. Our method could be generalized to other cases as well.


Introduction
Professor Roy Glauber made many ground-breaking contributions to physics; his creative use of quasi-classical probability distributions, quantum optics being a prime example. In the present paper, we show how to use Wigner distributions to go between different reference frames in general relativity, for example, between the Minkowski and Rindler frames. It is a pleasure to dedicate this article to Roy Glauber, physicist-friend-familyman.
In the 1970s, a relativistic quantum mechanics effect was predicted [1][2][3], where the particle content of a single quantum state in two reference frames was shown to be different. Specifically, a vacuum in an inertial frame is a thermal state to an accelerating observer. This is called the 'Unruh effect' or 'acceleration radiation' in the literature. The underlying reason for the effect is that particles correspond to excitations of positive frequency modes, and since different reference frames have different notions of time, observers in different reference frames do not agree on each other's definition of frequency .
Recently, quantum optics techniques were used for studying fundamental issues in quantum mechanics with relativity, quantum field theory in curved spacetime, and in particular, acceleration radiation and the dynamical Casimir effect [4][5][6][7][8].
In section 2, we introduce the basics of Rindler space [1,9] as related to reference frames of observers with constant acceleration; in section 3, we discuss the Unruh-Minkowski (UM) modes, which are the natural Minkowski positive frequency modes for translating between inertial and constant accelerating frames; in section 4 we review the Wigner distribution and its property that enables us to transform between reference frames; in section 5, we present and apply our method to the problem of transforming the Minkowski vacuum to an accelerating frame; in section 6, we use our method and obtain the Rindler space vacuum in terms of UM modes, and in section 7, we find the Minkowski number states in terms of Rindler modes. In appendix A, we show how the Minkowski vacuum state could be transformed into Rindler space variables without our method; in appendix B Original content from this work may be used under the terms of the Creative Commons Attribution 4.0 licence. Any further distribution of this work must maintain attribution to the author(s) and the title of the work, journal citation and DOI.
we give more information about Wigner distributions, and in appendix C, we explain how to obtain the Wigner distribution 5 for any quantum stater, and do the example of the vacuum state; in appendix D and E, we include a proof of the essential property of Wigner distributions which we employ; in appendix F, we discuss the parity operator, which is useful for understanding Wigner distributions; and in appendix G we derive the Wigner distribution of number states.
The particles that are the subject of this paper are massless scalar 'photons' in 1+1 spacetime, whose modes satisfy the Klein-Gordon equation. The treatment could be extended to the case of 3+1 spacetime. We choose treat only right-propagating modes; the left-propagating case being a straight-forward analog.

Rindler space basics
Minkowski space features two causally-independent wedges [1,9] (and beforehand, in a footnote in [10]), called the right and left Rindler wedges. See figure 1. Each Rindler wedge has its own set of coordinates, ( ) t z c , for the right wedge and for the left wedge 6 , where c is the speed of light. These are related to the Minkowski coordinates (ct, z) by [1,5,9] ( ) ℓ  e  c  ℓ   c  ℓ   z ct  ℓ   ℓ  z ct  z  ℓ  e  c  ℓ  ℓ   z ct  ℓ   z ct  ℓ   sinh  1  2  ln   cosh  1  2  ln  ,  1 ℓ ℓ or, in terms of the 'null' spacetime coordinates = - , and n t z = + c , the relationship between the spacetime coordinates is where ℓ is the length-scale in the problem such that the 4-acceleration of a classical object at 'rest' at the Rindler coordinate z = 0 is = a c ℓ 2 . To relate the coordinates in the right and left Rindler wedges, we take ⟶ℓ ℓbecause the constant acceleration trajectories of constant ζ worldlines in the right and left wedges accelerate in opposite directions. Equations (2) and (4) are two separate Rindler coordinates, each covering separate parts of the Minkowski space. Refs. [3,11] have shown that for each Rindler frequency | | x p W = c ℓ 2 , there are two positive frequency, right-moving Unruh-Minkowski (UM) modes, ( ) f x u and ( ) f x -u , and two positive-frequency left-moving UM modes, ( ) which are the subject of the following section.

The Unruh-Minkowski (UM) modes
Using the null coordinates =u ct z and = + v ct z as above, the wave equation for massless fields is f ¶ ¶ = 0 Since light signals have a 45°trajectory, no light signal can be sent between the right and left Rindler wedges-they are therefore said to be 'causally-disconnected.' The left and right Rindler wedges have independent particle excitations, corresponding to the annihilation operatorsˆx b R andˆx b L respectively. and the left-moving modes are ( ) f x v . We will not treat the leftmoving modes here because any results for the right-moving modes can be straight-forwardly generalized to the left-moving ones. We use the infinitesimal 'λ' to indicate that ( ) f x u is defined with a branch cut in the top-half complex u plane (correspondingly, ( ) f x u * has a branch cut in the bottom-half complex u plane). To an accelerating observer, these modes appear to have a frequency | | x p W = c ℓ 2 . That is, along the path z = 0, the modes go as ). In contrast, their complex-conjugates, , have negative norm. The mode ( ) f x u has a constant amplitude for positive u and a different constant amplitude for negative u; for positive ξ, it has a larger amplitude in the right wedge than in the left wedge, and vice-versa for negative ξ. The state defined byˆ| ñ = x a 0 0 for all ξ is the Minkowski vacuum state. The UM modes form a complete and orthogonal set of functions like the complex exponentials, and therefore we can expand the quantum fieldF in them left moving . 6

*
The fieldF could also be expanded in Rindler modeŝˆ[ where the subscripts 'R' and 'L' correspond to the left and right Rindler wedges, and left-wedge coordinates are primed, as above. As can be seen from the coordinate transformation equations (2) and (4), the mode , called 'the Rindler mode in the right wedge,' is only defined in the right Rindler wedge, and is a mode with a single Rindler frequency x p W = c ℓ 2 . Since the right and left Rindler wedges are causally-disconnected, there is no coherence between a right and left Rindler mode, and therefore this mode is zero in the left Rindler wedge. Similarly, the Rindler mode in the left wedge, , is zero in the right Rindler wedge. Comparing equations (6) and (7) for the fieldF, using equations (2) and (4) to relate the Minkowski null coordinate u to the right and left wedge Rindler null coordinates μ and m¢, one can see that for positive ξ, theâ andb operators are related bŷˆ( for all ξ. Throughout the paper, we focus on a single positive ξ. That is, we focus on the positive-norm UM modes f x and f x -, and their corresponding annihilation operatorsˆx a andˆxa , respectively.

Wigner distribution of a quantum state
Any quantum state could be represented by its Wigner dis- , which is a phase-space representation of its density matrix, and therefore, could be used for describing the quantum state and for calculating expectation values [12][13][14]. The Wigner distribution is a pseudo-probability function in phase-space (the term 'pseudo' is used because the Wigner distribution can be negative). The Wigner distribution ( ) w q p , , can be written as the expectation value of the displaced parity operator (ˆˆ) w q p , q p , [15], which is 7 [16] (ˆˆ) ( ) (see appendix B and F about the parity operator). That is, the Wigner distribution is the expectation value , where the factor of 2π appears for normalization. The c-numbers q and p correspond to the displacement of the configuration and the associated momentum operator,q andp. Instead of the configuration-conjugate momentum basis, one can generalize the Wigner distribution to other bases having a continuous spectrum [17]. For example, one could use the . To getw, one should first get the proper seed operator,ŵ , with the Wigner distribution , 2 i , * * being its expectation value. As we discuss in appendix D,ŵ is obtained from the configuration-conjugate momentum Wigner seed operatorŵ in equation (9) by replacing the configuration and conjugate momentum operators,q andp, and their corresponding c-functions, q and p, by the ladder operators,â andˆ † ia , and their corresponding c-functions, α and a i *, respectively. That is,ˆ(ˆˆ) where the c-number α corresponds to the displacement of the coherent state |añ, and the factor of i multiplying the α * is included so that the commutation relation ofâ withˆ † ia be the same as ofq withp [18, 19 16]. In appendix D we discuss that the Wigner distributions ( ) w q p , and ( ) a ã w i , * are related by 7 The semicolon ';' in the exponent denotes Schwinger operator ordering, where (in this case) all factors of (ˆ)p p are to the left of any factor of (ˆ)q q , as shown explicitly in equation (C3). direct substitution in their arguments, was obtained from equation (10).
Instead of treating the configuration and conjugate momentum as primary, we will treat the coherent-state basis as primary, and write the argument of the Wigner distribution as ( ) a W , which stands for ( ) a a W , * , which really stands for ( ) a ã w i , * . The Wigner seed operatorˆa W for a (single mode of the) field is the same as in equation (12), ; * and the Wigner distribution is its expectation value, As we mentioned in the end of the previous section, we focus on a single positive ξ for the Rindler and UM modes. The Wigner distribution 8 for the two modes f x and f x -is obtained from the density matrixr via 9 whereˆa W j is the Wigner seed operator of the j-th mode with parameter a j and corresponding creation and annihilation operatorsˆ( We show that describing a quantum state in terms of its Wigner distribution is very useful because it allows us to transform quantum states between different reference frames in a simple-minded way. In particular, our method entails only direct substitution in the arguments. The form of the transformation [see equations (8) and (19)] is obtained from the spacetime structure of the modes [11], whereas the transformation is performed in the coherent state basis.

The Minkowski vacuum
An important example is the Wigner distribution of the where a x is the phase-space displacement (in complex α-space) of the annihilation operatorˆx a or of its eigenfunction |a ñ x . Throughout the paper, we focus on a single positive ξ, keeping in mind that the full state contains all frequencies. For a single UM frequency 10 (that is, the ξ term in the product (17)), the Wigner distribution of the Minkowski vacuum state is Since the coherent state |añ has an average number of excitations | | a 2 , the value of ( ) a a x x -W , , after integrating over the phases of a x and a x -, heuristically corresponds to the probability of having | | a x 2 excitations of the mode f x and | | a x -2 excitations of the mode f x -. This state is given in terms of the coherent state a x and a x -of the UM modes, f x and f x -, respectively. As we discuss in appendix E, we can express the same quantum state (the Minkowski vacuum) in terms of coherent states of Rindler modes, b xR and b xL , of the right and left wedges, respectively; to do so, we make the canonical transformation in the Wigner distribution [14,16,[19][20][21][22] in equation (18),   If our observable is confined to the right Rindler wedge, we may integrate the Minkowski vacuum state, equation (20), over the b xL -phase space (because b xL refers to the left 8 See appendix C for more information about Wigner distributions and how they could be obtained from the density matrix. 9 We often write expressions like ( ) a a Rindler wedge modes). This would give the Minkowski vacuum state as it pertains to any observer or observable confined to the right Rindler wedge (in density matrix language, this amounts to performing a partial-trace over the left wedge variables). Doing so, we obtain the state which is a thermal state [20,21] with temperature , which is the Unruh result [3], where k B is Boltzmann's constant.

Rindler vacuum in Minkowski phase-space
One may also wish to transform a quantum state that is known in terms of Rindler excitations into Minkowski space. Whereas in the previous section, we showed how to transform a state known in Minkowski space into a Rindler space description, here we show how to accomplish the inverse transformation with our method.
In terms of Rindler modes, the Rindler vacuum statê From equation (8), we have that the annihilation operators for the Rindler modes in the right and left wedges are related to the UM modes by [11] ( ) and using their c-number analogs,

Minkowski number states
It is common perception that the Minkowski number states are tedious to treat, and indeed, we agree that they are laborious in the Rindler Fock space representation. However, in the Wigner phase-space language, a simplification occurs.
In terms of Unruh-Minkowski modes, a Minkowski mode number state is where L n is the Laguerre polynomial of order n, see appendix G. Using the transformation equations (19), we get that the n-Minkowski particle state is in terms of Rindler modes, where ( ) W 0 is the Minkowski vacuum state in terms of Rindler modes, equation (20).

Conclusions
We have presented a method for transforming any arbitrary quantum state between Unruh-Minkowski and Rindler modes, and vice-versa. This method could be applied to more general transformation cases. We have shown that translating the Minkowski vacuums and number states into the accelerating frame and the Rindler vacuum into the Minkowski frame are all easy to do using our method. vacuum in Rindler spacetime The Minkowski vacuum is defined as the state which is annihilated by all annihilation operators, for example Since | ñ 0 M must satisfy equation (A1) for every ξ independently, we see that the Minkowski vacuum is in terms of Rindler modes. Focusing first on a single ξ, we have that Then from equations (A1) and (A3), we have that and noticing thatˆ| (by definition of vacuum), and we find that equation (A4) implies thatˆˆ( giving thatˆx f is 11 Thus, in terms of Rindler modes, the Minkowski vacuum | ñ 0 M is -x f 1 exp is a normalization constant. Equation (A8) is just for a single ξ (notice that the normalization constant is ξ-dependent). For all ξʼs, we have in terms of Rindler Fock quanta, the Minkowski vacuum state is where | ñ n n , R L is the state of n R excitations in the right wedge and n L excitations in the left wedge. To get the state that an observer accelerating to the right would interact with, we need to trace-out the left-wedge (primed) quanta in the Minkowski vacuum density matrix Tracing over the left-wedge quanta, the density matrix is thermal

Appendix B. Position-momentum Wigner distributions
Here we discuss the fact that the position-momentum Wigner distribution is the expectation value of the seed operator [15,16] Starting with the more familiar formula for the Wigner distribution iq p and changing variables according to ⟶ ( which is the expectation value of the Wigner seed operator, equation (B1). Integrating the Wigner distribution over the configuration variable gives the probability of the associated momentum, and integrating over the momentum variable gives the probability of the configuration variable Therefore, the Wigner distribution satisfies the marginals (this is a property of all two-variable probability distributions). As implied by equation (B10), the Wigner distribution is normalized which is another property of probability distributions. However, the Wigner distribution can (and usually does) have negative values-so it is not a proper probability distribution. For these reasons, it is often referred-to as a quasi-probability distribution. 11 The sameˆx f is also obtained from the equationŝ Here, we give a quick introduction on how to obtain a Wigner distribution for a given quantum stater. The Wigner distribution ( ) a W of a quantum state is the expectation value [16,23], of the 'displaced parity operator' 12ˆa W [15], also known as the 'Wigner seed operator' [16] (ˆˆ) where the Wigner seed operator for mode f x is 13 ; * and similarly for the Wigner seed operator of the second mode,ˆa x -W . Example: Vacuum state. As an example of how to obtain the Wigner distribution from the Fock representation, we show how to obtain the Wigner distribution corresponding to the vacuum state | | ñá 0 0 ; * Since we have Schwinger ordering of the operator, we have Once we are familiar with Schwinger-ordered operators, we realize that we could have just written We give another example in appendix G, where we find the Wigner distribution of a number state.

Appendix D. Canonical transformations
Here we show that if , then the Wigner seed operators in theâ andb bases arê Our discussion is based on [20], where a more general case is done. An even more general case is done in [23]. As we learn from Royer [15], the Wigner seed operator is the 'displaced parity operator' (see appendix F) This implies that the commutators remain unchanged giving that ad bg -= 1. The transformation is also invertible,ˆˆˆˆ( Importantly, the displacement operator transforms likê from equation (D6) (having doffed the hats). A particular realization ofŜ iŝ 12 More on the parity operator in appendix F. 13 We sometimes express the operator's dependence onâ andˆ † a explicitly -as in equation (C5)-and sometimes not-like in equation (C4). Notice that the Wigner seed operatorˆ(ˆˆ) † a W a a , is a function of the field mode creation and annihilation operatorsâ andˆ † a , and is parameterized by a phasespace displacement α, which we include as a sub-script. At a = 0, for instance,Ŵ 0 is the displaced parity operator with zero displacement-that is, it is the parity operator-see appendix F. The Wigner distribution ( ) a W is a function of the phase-space displacement parameter α.
giving that equation (D4) iŝˆˆˆˆ( and therefore, operating from the right withˆ- which is a surprising result, and is true because , . From equations (D7) and (D11), we see that and thus, the Wigner distribution corresponding to the operatorˆ(ˆˆ) † where x and k are the c-numbers corresponding to eigenvalues of and, respectively. Alternatively, given

Appendix E. Two-mode Wigner identities
Here, we generalize the results of appendix D to the twomode case, which is interesting for us, because of equations (8) and (24). Generalizing equation (8), we have the transformationˆˆˆˆˆˆˆ( Our case (which is less general), could be obtained from a a a a exp 1 2 where the αʼs as a function of the βʼs are obtained from equation (E4) after replacing the operators by their c-numbers. We see that sinceŜ has dependence only on products of even powers of creation and annihilation operators, and therefore,ˆˆ( In terms of the definition in equation (D3), the Wigner seed operator iŝ From equations (E8) and (E11), we see that 14 In the body of the paper, e.g. in equations (17), (18), and (20) for any α, which shows that that the square ofŴ 0 is the identity operator