Vacuum thermal effects in flat space-time from conformal quantum mechanics

The generators of radial conformal symmetries in Minkowski space-time can be mapped to the generators of time evolution in conformal quantum mechanics. Within this correspondence we show that in conformal quantum mechanics the state associated to the inertial vacuum in Minkowski space-time has the structure of a thermofield double. Such state is built from a bipartite"vacuum state", the ground state of the generators of hyperbolic time evolution, which cover only part of the time domain. When time evolution is restricted to a finite time domain one obtains the temperature perceived by static diamond observers in the Minkowski vacuum. When time evolution is determined by dilations, covering only half of the time line, the temperature of the thermofield double corresponds to the non-vanishing temperature perceived by Milne observers whose proper time evolution is confined to the future cone (Milne universe) of Minkowski space-time. The two pictures are related by a conformal transformation on the real line. Our result provides a purely group theoretical derivation of the Milne and diamond temperature and shows that the fundamental ingredient for vacuum thermal effects is the presence of a horizon rather than acceleration.


INTRODUCTION
The notion of vacuum state in quantum field theory is observer dependent [1]. The importance of this basic fact was only started to be appreciated with the discovery of black hole quantum radiance [2,3], which is one of its main consequences. The existence of a temperature and thermal features associated to the horizon of a black hole is directly related to two of the major open issues in quantum gravity: the fate of unitarity in the quantum evolution of black holes [4][5][6] and the nature of their Bekenstein-Hawking entropy [7,8], and it is at the basis of the interpretation of * Electronic address: michele.arzano@na.infn.it gravity as a "thermodynamics of space-time" [9].
The ambiguity in the choice of the vacuum state in curved space-time relies on the possibility of having different choices of time-like Killing vectors that can be used to decompose the space of solutions of the classical field equation into positive and negative energy subspaces [10]. The former will define, together with a positive definite inner product, the one-particle Hilbert space for static observers with respect to the Killing vector which generates evolution in their proper time.
From such Hilbert space the standard Fock space construction follows with the associated vacuum and multi-particle states [11]. The Hawking effect boils down to the fact that the Hartle-Hawking vacuum state associated to freely falling observer is a thermal state at the Hawking temperature for static observers with respect to Schwarzschild time whose associated Killing vector becomes null at the horizon.
In maximally symmetric spaces the quantization ambiguity associated to different possible choices of time evolution is eliminated by the requirement of invariance of the vacuum state under the full group of isometries. In Minkowski space-time this singles out the standard vacuum state associated to inertial time evolution under the ordinary time translation Killing vector. However, as it is well known, one can also consider time evolution generated by a boost Killing vector which is time-like in the Rindler wedges and whose orbits correspond to worldlines of uniformly accelerating observers. For such observers the inertial vacuum is populated by a thermal distribution of particles at a temperature proportional to the magnitude of their four-acceleration: the so-called Unruh temperature [3].
Quite interestingly, if one restricts the attention to scale-invariant fields the range of possible candidates for generators of time evolution can be extended to include conformal Killing vectors.
In Minkowski space-time this has recently been used [12,13] (see also [14,15]) to consider "Milne quantization" of a two-dimensional massless scalar field in the future and past cones, analogous to Rindler quantization in the Rindler wedges, where time evolution is generated by the dilation conformal Killing vector. Observers along the radial orbits of dilations are comoving observers in a Milne universe (which is just Minkowski space-time described in expanding coordinates covering only the future cone) which perceive the inertial vacuum as populated by a thermal distribution of Milne particles.
These considerations suggest that in higher dimensions, and here we refer for simplicity to the 3 + 1 dimensional case, one could define time evolution in terms of conformal Killing vectors which are time-like in certain regions of Minkowski space-time and similar thermal effects should manifest. Of particular interest are, to this extent, radial conformal symmetries, whose full classi-fication for Minkowski space-time has been given in [16]. Radial conformal Killing vectors divide   Minkowski space-time into causal domains separated by light-like surfaces which are conformal Killing horizons, they are analogous to stationary spherically symmetric Killing horizons of black hole geometries and have similar thermodynamic properties [17,18]. Among such radial conformal Killing vectors are the ones which map a causal diamond into itself and whose orbits describe worldlines of observers confined within the diamond. The fact that these diamond observers should perceive the inertial Minkowski vacuum as a thermal state was first suggested in [19] where the claim was supported with arguments which used the so-called "thermal time hypothesis". Another attempt to provide evidence of such diamond temperature was given in [20] using a model of Unruh-DeWitt detector moving along the diamond observer trajectory.
In [21] the present author pointed out that radial conformal symmetries are in correspondence with generators of time evolution in conformal quantum mechanics, a 0 + 1 dimensional conformal field theory [22]. Using this correspondence, for which the "wordline quantum mechanics" of diamond observers at the origin 1 is viewed as a conformal quantum mechanics, it was shown that the Minkowski vacuum is perceived by diamond observers as a thermal state at a temperature inversely proportional to the size of the diamond. The results of [21], besides providing a simple but powerful tool for studying the thermodynamics of causal regions in Minkowski space-time, also show that the possibility of selecting different generators of time evolution in conformal quantum mechanics leads to thermal effects in complete analogy with higher dimensional quantum field theory. Since the focus of [21] was on causal diamonds, and the corresponding hyperbolic time evolution restricted to a finite time domain in conformal quantum mechanics, the natural question is whether similar thermal effects can be associated to other choices of time evolution like, for example, the one determined by the generator of dilations, and what is the physical interpretation of the associated temperature in Minkowski space-time.
In this work we show that, besides the "diamond temperature" associated to time evolution restricted to a finite time domain of conformal quantum mechanics, there is a natural temperature associated to time evolution generated by dilations and restricted to positive (or negative) times.
These two thermal effects have their origin in the thermofield double structure of the vacuum state associated to the "inertial" time evolution generator which is here evidenced using purely group theoretical arguments. The thermal effect associated to time evolution restricted to half of the time line indicates that in Minkowski space-time static Milne observers, i.e. observers at the origin whose proper time evolution is determined by dilations, perceive the Minkowski vacuum as a thermal state. Such result provides evidence for the existence of a "Milne temperature" for space-time dimensions higher than two and exhibits its intimate connection with diamond temperature given that, as it shown below, the two pictures are connected on the conformal quantum mechanics side by a conformal transformation. Finally, since both the diamond and Milne temperatures are nonvanishing along the worldlines of static observers sitting at the origin with vanishing acceleration, our result demonstrates that the fundamental ingredient for quantum vacuum thermal effects is the presence of a horizon, in this case associated to the non-eternal nature of the lifetime of the observer, rather than a non-vanishing acceleration.

RADIAL CONFORMAL KILLING VECTORS IN MINKOWSKI SPACE-TIME AND TIME EVOLUTION IN CONFORMAL QUANTUM MECHANICS
Let us start from the line element of Minkowski space-time written in spherical coordinates where dΩ 2 = dθ 2 + sin 2 θ dφ 2 . The most general radial conformal Killing vector ξ for which L ξ η µν ∝ η µν where L ξ is the Lie derivative and η µν is the Minkowski, metric is given by [16] where K 0 , D 0 and P 0 generate, respectively, special conformal transformations, dilations and time translations and close the sl(2, R) Lie algebra These generators can be written in terms of radial and time derivatives as so that (2) becomes and in light-cone coordinate u = t + r and v = t − r we have In conformal quantum mechanics the generators K 0 , D 0 and P 0 play the role of different time evolution operators. In terms of the inertial time variable τ associated to the generator P 0 they can be written as differential operators The first important observation is that radial conformal Killing vectors for static observers at the origin r = 0 and on the light-cones u = const. or v = const. coincide with the most general time evolution operator in conformal quantum mechanics The causal structure of the Killing vector (7) in Minkowski space-time has been studied in [16].  [12,15,16]. The other relevant generator is the radial conformal Killing vector which maps a causal diamond, i.e. the region |t| + |r| < α, into itself given by The causal structure of this Killing vector for which ∆ > 1 and a = 0 is more articulated. It is null on the light-cones emanating from r = 0 at times α and −α. In order to do so let us first introduce time coordinates adapted to the evolution defined by the generators D 0 and S 0 i.e. such that defining the Milne and diamond time for static observers with semi-infinite and finite lifetimes respectively. We can easily derive the transformation between the inertial time τ and the Milne and diamond time coordinates ν and σ comparing (14) and (15) with the expressions (10) and (13). One obtains where the integration constants C 1 and C 2 can be determined by requiring dimensional consistency.
We set the reason for the choice 2α for the argument of the log will be clear later. We thus have for the Milne time We see that on the positive half-line τ > 0 as ν ranges from −∞ to ∞ we go from τ = 0 to τ = ∞.
On the negative half line the time flow is inverted and we go from τ = 0 to τ = −∞ as ν ranges from −∞ to ∞. Notice that, even though there is no natural time scale associated with the future cone, a constant with dimensions of time is needed in order to define a time evolution operator in terms of the generator of dilations τ ∂ τ . This time scale is the point on the real line measured in time τ corresponding to the origin ν = 0 of Milne time.
For the S-time variable σ we have that in the region |τ | < α, i.e. within the diamond, i.e.
while for |τ | > α one has We see that within the diamond as the diamond time ranges from −∞ to ∞ we cover the region from t = α to t = −α.
Let us now consider a new time coordinate τ ′ (τ ) in terms of which the dilation generator D = τ ∂ τ acts as the generator S of conformal transformations which preserve the diamond, i.e.
For dimensional consistency we choose for the integration constant C = − log A, where A has dimensions of time and thus must be proportional to α. Restricting to the positive half-line τ > 0 we have In order to determine the constant A we first notice that the point τ ′ = −α gets mapped to τ = 0 while τ = α to τ = ∞. The origin of the diamond τ ′ = 0 gets mapped to τ = −A so we know that A must be negative.
We thus have which can be inverted to give Notice how this transformations correspond to the map from the causal diamond to the Rindler wedge and viceversa [25,26] when written in radial light-cone coordinates, indeed the Rindler wedge and the future cone meet on the light cone.

THE THERMOFIELD DOUBLE OF CONFORMAL QUANTUM MECHANICS
As shown in [27,28] Let us look for states |τ on which the SL(2, R) generator H acts as a τ -derivative These states were first constructed in [22] starting from an irreducible representation of the sl(2, R) Lie algebra. Upon defining the operators and with commutation relations such representation is realized in terms of the eigenstates |n of L 0 L 0 |n = r n |n (35a) r n = r 0 + n, r 0 > 0, n = 0, 1 . . .
where r 0 is related to the eigenvalue of the Casimir operator of the sl(2, R) algebra In [27,28] it was found that the |τ states can be obtained from the |n = 0 "vacuum state" via where The action of the other generators on such τ -states are given by In [27] the two point functions of the theory are identified with the inner product between the τ -states where r 0 plays the role of the conformal weight. General n-point functions with n > 2 are obtained by inserting generic primary operators between τ 1 | and |τ 2 . When τ = 0 one has ω(τ ) = 1 and thus |τ = 0 = Γ(2r 0 ) 1/2 exp(−L + ) |n = 0 (42) One immediately sees from (41) that such τ -vacuum is not normalizable due to the divergence of the two-point function for coincident points.
In [21] it was noted that for r 0 = 1 the two-point function (41) is proportional to the two-point function of a free massless scalar field in Minkowski space-time evaluated along the trajectory of a static inertial observer at the origin. This fact was used to derive the diamond temperature 3 upon re-writing (41) in terms of the time variable associated to the diamond time evolution generator S = iS 0 . Here we provide an alternative derivation of the same temperature by looking directly at the structure of the τ -vacuum. This will lead to a unified description of the diamond and Milne temperatures.
Let us start from a simple observation which, nonetheless, is a key passage of our derivation. It is known from the quantum optics literature (see e.g. [30,31]) that the operators L ± and L 0 with commutation rules (34) can be realized in terms of two sets of creation and annihilation operators a L , a † L , a R , a † R as It immediately follows from (42) that the τ -vacuum can be written as 4 where we made explicit the bipartite structure of the vacuum state |n = 0 emerging from the decomposition (44) With simple manipulations we have that where |n L and |n R are eigenstates of the "left" and "right" number operators N L = a † L a L and N R = a † R a R . Our second key observation is that the Lie algebra can be realized via another combination of H, D and K, namely connection is a bit more subtle and should be understood in terms of a state-operator correspondence illustrated in [27]. As observed in [21] the Minkowski light-cone two-point function maps to the CF T 1 two-point function constructed from the |τ states (41). The latter can be written as Indeed it turns out [27] that such two-point functions can be expressed as Such superposition of states is the same as the one appearing in the thermofield double state of a bosonic oscillator [32,33]. Indeed, starting from the τ -vacuum density matrix ρ = |τ = 0 τ = 0| and tracing over one set of L or R degrees of freedom one obtains a thermal density matrix at temperature 1/2π with respect to the modular Hamiltonian S. Now, of course, S does not have the dimensions of a Hamiltonian, rather one should see the operator S α as the modular Hamiltonian of the system so that the temperature associated to the thermofield double (52) is given by i.e. precisely the diamond temperature obtained in [21].
An interesting upshot of the analysis above is that it can be used to also give a group theoretical derivation of the Milne temperature. Indeed, under the conformal map (29) the identification (48) leads to the following realization of the algebra (47) With such identification the vacuum state |n = 0 is now seen as the conformal quantum mechanics analogue of the vacuum state associated to D, the generator of conformal time translations in the future cone, and the temperature is simply the temperature perceived by a Milne observer sitting at the origin of Minkowski spacetime in the inertial vacuum. In two space-time dimensions, where null planes and null cones coincide and Milne and Rindler particles can de identified on the the light cone [12], our arguments give a unified, group theoretical derivation of the diamond, Milne and Unruh temperatures.

CONCLUSIONS
The considerations above show that the structure of conformal quantum mechanics is rich enough to reproduce, in the extremely simplified context of a one-dimensional model, the basic features which characterize vacuum thermal effects in quantum field theory, i.e. the appearance of the global vacuum as a thermal state populated of excitations associated to Hamiltonians whose evolution does not cover the entire time domain of the theory. Remarkably, using the correspondence between time evolution in conformal quantum mechanics and radial conformal symmetries in Minkowski space-time seen as generators of proper time, this result provides direct evidence for the fact that the inertial vacuum appears as a thermal state to observer whose time evolution is not eternal. The results presented suggest that the thermodynamic properties of causal diamonds and the Milne "patch" of Minkowski space are intimately connected. It is tempting to speculate that the tools of conformal quantum mechanics which are used to provide evidence for such connection might provide an alternative and perhaps simpler way to describe entanglement features of quantum fields across the causal boundaries of such regions. A thorough exploration of such hypothesis is left to future studies.