Physics Letters B 781 (2018) 611–615

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Physics Letters B

www.elsevier.com/locate/physletb

Unruh effect as a result of quantization of ∗ Lucas C. Céleri, Vasileios I. Kiosses

Instituto de Física, Universidade Federal de Goiás, Caixa Postal 131, 74001-970, Goiânia, Brazil a r t i c l e i n f o a b s t r a c t

Article history: A way to encode acceleration directly into fields has recently being proposed, thus establishing a new Received 26 March 2018 kind of fields, the accelerated fields. The definition of accelerated fields points to the quantization of Received in revised form 24 April 2018 and time, analogously to the way quantities like and momentum are quantized in usual Accepted 24 April 2018 quantum field theories. Unruh effect has been studied in connection with quantum field theory in Available online 14 June 2018 curved spacetime and it is described by recruiting a uniformly accelerated observer. In this work, as Editor: N. Lambert a first attempt to demonstrate the utility of accelerated fields, we present an alternative way to derive Unruh effect. We show, by studying quantum field theory on quantum spacetime, that Unruh effect can be obtained without changing the reference frame. Thus, in the framework of accelerated fields, the observational confirmation of Unruh effect could be assigned to the existence of quantum properties of spacetime. © 2018 The Authors. Published by Elsevier B.V. This is an open access article under the CC BY license (http://creativecommons.org/licenses/by/4.0/). Funded by SCOAP3.

1. Introduction spacetime must be quantized [6]. String theory, in a different set, eventually came to the same conclusion [7]. While quantum mechanics arose from experimental observa- Recently, a radical different approach was proposed by the au- tions which could not be reconciled with classical physics, till now, thors, leading to the quantization of space and time by considering there is no experimental fact indicating that space and time are not gravity but acceleration [8]. As is well known, Einstein initially quantized. This is because the standard model, which describes used accelerated systems as a guide to understand the nature of all matter we have observed, is based on flat space quantum field gravity in classical physics. We believe this can also be done for theory and , which describes gravity, takes no ac- quantum theory. In this vein, accelerated fields were introduced. count of quantum theory. The construction of accelerated fields differs from ordinary quan- In general relativity, the gravitational field and the spacetime tum field theory and provides a mathematically consistent way to are the same physical entity. All fields we know exhibit quan- quantize spacetime. In this approach, space and time are quantized tum properties at some scale, thus we believe that spacetime must in the way quantities like energy and momentum are quantized in have quantum properties as well. The idea of quantizing spacetime ordinary quantum field theories. dates back to the early days of quantum theory as a way to elim- The theory of accelerated fields, as presented at [8], is an effort inate infinities from quantum field theories [1,2]or just to survey to quantize spacetime, while it is kept separated from the mat- the consequences of this assumption [3]. ter fields. Strictly speaking, the aim was only to locally describe The first argument suggesting that due to gravitational field, the quantum behavior of the gravitational field. In this article, we spacetime breaks down at very short distances was advanced by proceed a step further and investigate the effects of accelerated Bronstein [4]. Today, while it is generally accepted that space- fields on the Klein–Gordon massive field, by defining accelerated time is quantized, there is disagreement as to how quantization Klein–Gordon massive fields and comparing them with the in- manifests itself [5]. Currently, the main candidates for a theory ertial ones. As we will see below, the definition of accelerated of are and string theory. Klein–Gordon fields actually gives rise to creation and annihilation Loop quantum gravity was the first major theory to postulate that operators that encapsulate noninertial effects. Spacetime becomes dynamical and affects the matter fields defined on it. The new cre- ation and annihilation operators defined by this theory imply a modification of the usual commutation relation. Specifically, the * Corresponding author. E-mail addresses: lucas @chibebe .org (L.C. Céleri), kiosses .vas @gmail .com new commutation relation is divided in two parts, the first one (V.I. Kiosses). corresponding to the standard commutation relation while the sec- https://doi.org/10.1016/j.physletb.2018.04.050 0370-2693/© 2018 The Authors. Published by Elsevier B.V. This is an open access article under the CC BY license (http://creativecommons.org/licenses/by/4.0/). Funded by SCOAP3. 612 L.C. Céleri, V.I. Kiosses / Physics Letters B 781 (2018) 611–615 ond one reflects the quantum nature of spacetime. We thus find the definition of the positive- and negative-frequency solutions lies  that space and time, due to acceleration, ought to be described in the existence of a space-like Killing vector field, ∂p , in momen- as a non-commutative quantity. There has been much work re- tum space [8]. cently studying consequences of formulating quantum mechanics Working in the space spanned by the positive frequency modes  on non-commutative manifolds in an attempt to describe quantum ut , one can define the field operator Ψ in the language of the sec- gravity [9,10]. In this work we show that acceleration can be em- ond quantization in the usual way ployed to develop theories of deformed commutation algebras as    well. † ∗ Ψ(pμ) = dt a u (pμ) +a u (pμ) . (2) Moreover, considering the state of the accelerated t t t t Klein–Gordon field we find that it is equivalent to an inertial  † Klein–Gordon field in thermal equilibrium at T . In at (at ) is the annihilation (creation) operator, which acts on co- other words, we find that in the presence of the accelerated field, ordinate space and annihilates (creates) excitations at space–time an inertial observer should detect a flux of thermal particles whose point (t, xt ). From these operators, the field Hamiltonian takes the temperature is proportional to the acceleration defined by the ac- form celerated field.  dt The primary significance of Unruh effect [11]lies in the ar-  = √ †  H xt at at . (3) gument that the very concept of particles is frame-dependent, 4π xt demonstrating that the key features of Hawking exists The Hamiltonian was defined in the following way. First we con- without gravity. Here we show that it is possible to derive Un- struct the Lagrange density associated with equation (1)by invert- ruh effect without changing the reference frame, by just studying ing the Euler–Lagrange equation. Then, Noether’s theorem provides quantum field theory in the presence of the accelerated field. In us the momentum independent quantity that plays the role of the this way, Unruh effect acquires its own source, responsible for the Hamiltonian. Finally, by using Eq. (2)we can rewrite the Hamilto- observable particle creation rate, in the same spirit as Hawking ef- nian in the form shown in Eq. (3). fect has the gravitational field in the form of Black holes [12,13]. By defining the conjugate momentum as Π (p) =∂ Ψ (p), and We observe that Unruh and Wald [14]made clear that Unruh ef- E p E postulating the canonical equal-momentum commutation relations fect is compulsory for the consistency of field theory in accelerated frames. The main result of this paper is that Unruh effect is nec-       Ψp(E), Ψp(E ) = Πp(E), Πp(E ) = 0, essary in order to describe the consequences of formulation field     theories on quantum space. Ψp(E), Πp(E ) = i δ(E − E ), It is interesting to note that in the context of string theory there are some works on quantum space which address Unruh effect we get [15–17]. In these works, the authors were interested in defining † † †  a a  = a a  = a a  = − a quantum system with a built-in length scale. A quantum space [ t , t ] t , t 0and t , t δ(t t ). (4) were defined as a choice of polarization for the Heisenberg algebra of quantum theory. The derived quantum space employed in the Since the individual solutions ut are associated with positive present work, on the other hand, is based on Schrödinger quan- frequency, and the Hamiltonian is a sum over the contributions tum mechanics and relies on the classical notion of momentum from each t value, there will be a single vacuum state |0a, defined  |  = †|  appearing as arguments of the wave function [8]. by at 0a 0. The excitations will then be defined by at 0a and In [8]it has been shown a way to build particle states charac- are interpreted as single accelerated particle. In the usual quantum terized by uniform acceleration. In order to set notation and the field theory every excitation is said to carry a specific energy and necessary mathematical background, in the following we briefly momentum. In our case, since the wave equation is defined in the summarize the main elements of this structure. After this, we momentum space, we say that every excitation will carry a specific proceed on its generalization by including states which describes t and xt . accelerated Klein–Gordon particles, and state our main result. We consider quantum field theories in two dimensions with metric 3. Results signature +−. Furthermore, we shall use units such that h¯ = c = = k 1, unless specified otherwise. Having defined the accelerated fields in momentum space, we return now to coordinate space in order to discuss the ordinary 2. Background theory of quantum fields in the presence of an accelerated quan- tum field. We consider the simplest possible case, a real mas- Let us start by considering a real scalar accelerated field Ψ(pμ) sive scalar field  in –time obeying the massive in momentum space, defined by the wave equation [8] Klein–Gordon equation (∂2 − ∂2 + m2)(t, x) = 0. The normal-   t x 1 mode solutions to this equation, in Minkowski coordinates, are 2 2  μ − − ∂ − ∂ − Ψ(p ) = 0, (1) μ = i(E pt px) = 2 + 2 E p α2 φp(x ) e / 4π E p , with E p p m . The Klein– Gordon inner product, which is expressed as an integral over a μ = with p (E, p) being the four momentum while α represents constant-time hypersurface, allow us to define the annihilation op- the proper acceleration. From here on, quantities denoted with erator associated to φp by ap = φp,  . From this we can write (without) a tilde are associated with Eq. (1)(the Klein–Gordon KG    equation). There is a clear similarity between Eq. (1) and the = + † ∗ Klein–Gordon equation. However there is also an important dif- (t, x) dp apφp(t, x) apφp(t, x) , ference: For accelerated fields the wave equation is space-like. The |  |  = way to deal with this discrepancy is to solve the theory by intro- whose associated vacuum state 0I is defined by ap 0I 0. The  μ = † |  ducing an√ orthonormal set of mode solutions given by ut (p ) states ap 0I are interpreted as single particle states with momen- i(Et−pxt ) 2 2 e / 4π xt , with xt ≡ x(t) = t + 1/α , and claiming that tum p and energy E p , L.C. Céleri, V.I. Kiosses / Physics Letters B 781 (2018) 611–615 613

|  { ∗} A Computing the action of (t, x) on 0I one finds the set of modes φp, φp x→xt . In the standard way, ΨKG(t) is  represented as an operator-valued distribution in a Fock space ∗ † |  A |  = | = |  spanned by the vacuum state 0A , defined by ap 0A 0 and by (t, x) 0I dp φ (t, x)ap 0I   p † † the (unnormalized) n-particle states of the form aA aA ...   p1 p2 † which corresponds to the superposition of single particle momen- A |  ap 0A . tum eigenstates and thus, corresponds to a particle at (t, x). j   A † |  2 + 2 The theory of accelerated fields, as we have seen, has been es- The excitations defined by ap 0A carries energy p m A |  tablished in momentum space, thus the basic degrees of freedom while the ones associated with ΨKG(t) 0A are located at position are operator valued functions of energy and momentum, i.e. the t2 + 1/α2. Thus, the field operator is associated with single par-  μ  μ field operator Ψ(p ) and its conjugate momentum Π(p ). However, ticles with mass m and acceleration α. In this space the (renormal- writing the wave equation (1)in Fourier space (which actually is ized) Hamiltonian is given by the coordinate space), each Fourier mode of the field is treated as    † A = A A an independent oscillator with its own annihilation and creation H dp E p ap ap operator. Hence, from Eq. (2)the Fourier transform of the field  μ  Ψ(p ) reads = A † A where the operator N A ap ap counts the number of acceler- ated particles with momentum p.  =   μ +†∗ μ Ψ(t) at ut (p ) at ut (p ). The zero-point fluctuations that a quantum harmonic oscillator

† exhibits arise formally from the non-commutativity of the cor- The operator a (a ) creates (annihilates) particles associated with t t responding creation and annihilation operators. Like a harmonic the mode functions u (pμ). For a given time t, there are two plane t oscillator, if we consider the field component wave solutions for ut , one with positive xt and one with negative † ∗ xt . This is similar to the solutions for the Klein–Gordon equation A =  μ +  μ ΨKG(t) t ut (p ) −t u−t (p ), for a specific momentum, composed of positive and negative en- |  ergies. Therefore, the set of solutions of the Eq. (1)are distributed which has zero mean in the vacuum state 0I , it must undergoes into two space-like separated regions. zero point fluctuations due to the non-commutativity of t and †   , Considering the Klein–Gordon field operator (t, x), we impose t  the equal-time canonical commutation relation (t, x), (t, x ) =    A † A μ 2 1 iδ(x − x ), with (t, x ) being the momentum conjugate to (t, x). 0I | Ψ (t) Ψ (t)|0I =|u−t (p )| = . (7) KG KG 4π x Moreover, we add the relation t The meaning of this equation is that, at every point in momen-   (t, x), †(t , x) = δ(t − t ). tum space, characterized by a specific acceleration (xt is accelera- tion dependent), a pair of spatially separated accelerated particles The algebra stemmed from the above commutation relation, comes out in coordinate space. This is similar to the usual quan- typical for creation and annihilation operators, are identical to tum field theory, when vacuum fluctuations induce the creation of those we found for the quantized accelerated field by using plane a time-like pair. We found that the relationship between ΨA (t) and (t, x) is waves as an expansion basis. Thus, writing KG given by (5). Since both fields were quantized using the normal    † ∗ Minkowski modes, their difference reduces to the difference be- ξ A (pμ) = dt  u (pμ) +  u (pμ) , KG t t t t tween their corresponding creation and annihilation operators. In the case of real-valued Klein–Gordon fields, i.e. Hermitean field † † the operators t and t acquire a dual role. As Fourier coeffi- operators  = , the Klein–Gordon inner product of the field op- A A ∗ cients in the expansion of the wave operator ξKG they create and erators Ψ (t) and φ (t, x) (φ (t, x)) gives the Fourier coefficient  KG p p annihilate particles associated with the mode function ut and, as a † aA ( aA ), and, due to Eq. (5), we have Klein–Gordon field operator, they create Klein–Gordon particles at p p A μ  ∗   ∗  (t, xt ). Note that ξKG(p ) is a combination of quantities with and u +u u +u A ∗ t t † ∗ t t i2E pt without the tilde. a = ap u + + a− u − e (8) p t 2 p t 2 Now, the field where the wave functions ut are taken at E = E p . The commuta- A μ † ∗ μ A A † Ψ (t) = tut (p ) +  u (p ). (5) tion relations for ap and (ap ) can be found to be KG t t   2 + ∗ 2 obeys the Klein–Gordon equation. Since we are in Minkowski A A † † ut (ut ) [a ,(a  ) ]=2[ap,a  ] 1 + . (9) spacetime, the time-translation symmetry allows us to unambigu- p p p 2 ously define a set of positive- and negative-frequency modes, − whose temporal behavior is given by e iEpt and eiEpt , respectively, As we see, the presence of the accelerated wave modes in the and to expand the field operator Ψ in terms of these modes. Choos- solution of the Klein–Gordon equation modifies the commutation { ∗} relation between creation and annihilation operators. ing the pair φp, φp , which are expressed in Minkowski coordi- nates, the expansion of Ψ in terms of these modes reads The transformation (8), since it mixes annihilation and creation operators, is actually the Bogolubov transformation. Thus, the vac-      † uum states defined by the inertial annihilation operator a and the A = A + A ∗ p ΨKG(t) dp ap φp(t, xt ) ap φp(t, xt ) . (6) A accelerated annihilation operator ap are distinct due to the pres-   ence of ut in the latter. More specific, the expectation value of the A † A with the operator coefficients, ap and ap , to be interpreted, number operator N A for the of the inertial free the- as usual, as creation and annihilation operators with respect to ory |0I  is given by 614 L.C. Céleri, V.I. Kiosses / Physics Letters B 781 (2018) 611–615   1 sin2[(E t − px )]  | | = −2 − ∗ 2 = p t 4. Conclusions 0I N A 0I 1 ut (ut ) , 2 4π xt which does not vanish for every p. Particle creation in the case of Hawking effect for black holes Until now we have shown the effect of accelerated fields on the and Unruh effect for accelerated observers are consequences of the commutation relations of an inertial field theory and the disagree- application of the general framework of quantum field theories in ment between an inertial- and an accelerated Klein–Gordon fields curved . Since, in a curved spacetime setting, there is on the definition of the ground state. Now we will show that this no analog of the preferred Minkowski vacuum, the very notion of difference is due to the thermal behavior of the accelerated field. particles becomes ambiguous. Our formulation avoids such diffi- In order to show this, we derive a relation between the inertial de- culties because it does not rely on these ideas since we work in scription of an inertial- and an accelerated Klein–Gordon field. The flat spacetime and considering only inertial reference frames. In approach presented below is based on Ref. [18] where the thermal our case, particles will be emitted from the vacuum of an inertial effect of acceleration was computed by means of the field correla- quantum field due to the presence of the accelerated field, that tion function. acts as a source for such effect. We work with the inertial Klein Gordon field (t, x) and the Keeping in mind that the mathematical language we use here A accelerated one ΨKG(t), as defined in the previous section, with the is that of quantum field theory and not that of string theory, our only difference being that here it is assumed to be massless, in formulation integrates some of the ideas developed in metastring order to simplify the calculations. theory [19]. The habitat of metastrings is a form of quantum space- Starting with (t, x), consider the two-time correlation func- time which reveals a geometric structure behind quantum theory. tion (t, 0) (t + τ , 0) at a specific point in space for the field In that structure spacetime and momentum space appear to hold in thermal equilibrium at temperature T , i.e. the mode whose fre- equal parts. Sharing the same view, we have promoted momen- quency is E p has n(E p ) particles on average tum space by constructing a relativistic field theory in momentum    − space analogously to standard field theories in spacetime.  1 † = − = E p /T − The concept of Born reciprocity [20]simplystates that physics ap ap δ(p p)n(E p), n(E p) e 1 . should be equivalently formulated from the position and momen- From this it follows that tum point of views. In metastring theory, Born reciprocity is re- 1 lated to a fundamental symmetry of string theory, that of T-duality, (t, 0) (t + τ , 0) =− (π T )2csch2 (π T τ ) (10) π which is generalized to Born duality [19]. In our formulation T-du- A ality reduces to a Fourier transform, which exchanges field op-  We proceed with ΨKG(t), considering the correlation function A A erators in momentum space with field operators in spacetime. Ψ (t1) Ψ (t2) in the vacuum state |0A . In this case it holds  KG  KG  0A      Precisely this existence of two types of quantum field theories in A A = A † A † = A † A = A A † = ap a  (ap ) (a  ) (ap ) a  0 and ap (a  ) spacetime (the standard field theory and the Fourier transform of p 0 p 0 p 0 p 0  A A A A δ(p − p ). Therefore, from Eq. (6), we obtain the accelerated field theory), interacting in the way presented in the present work, results in Unruh effect. While the first one, the   1 1 ΨA (t ) ΨA (t ) = , standard field theory, is associated to the dynamics, the second one KG 1 KG 2 0 2 2 A π x − t should be related to kinematics. In that case the vacuum is iden- = − = − with t t2 t1 and x xt2 xt1 . By construction, the relation tified with a state that is annihilated by both the “dynamical” and 2 − 2 = 2 xt t 1/α holds. As this represents hyperbolic curves, we can “kinematical” annihilation operators. This is reminiscent of the Un- use hyperbolic functions and set ruh effect where different vacua correspond to different reference frames. The analogy is established by identifying the accelerated 1 = observer’s reference frame with the inertial one in the presence of xti cosh(αρi) (11) α accelerated fields i.e. the “kinematical” fields. 1 Even though gravity, as described in general relativity, appears ti = sinh(αρi), i = 1, 2 (12) α not to be compatible with quantum field theory, this is not the 2 2 with ρi a parameter. One can calculate the difference x − t case for the accelerated fields. The fact that the accelerated fields by making use of Eqs. (11) and (12) are quantized under the canonical procedure allowed us to pre-   cisely calculate the effect of acceleration on the inertial wave op- 4 α(ρ2 − ρ1) x2 − t2 =− sinh2 . erator. In addition, we went one step further and argued that this α2 2 change of the wave operator, as measured by an inertial observer,   A A is responsible for the Unruh effect. The correlation function Ψ (t1) Ψ (t2) in the vacuum of the KG KG 0A In this work we adopt the view that certain classical notions accelerated Klein–Gordon field, as measured by an inertial ob- about space and time are inadequate at the fundamental level. server, is given by   We define accelerated Klein–Gordon fields by forming a notion of   2 − A A α 2 α(ρ2 ρ1) Klein–Gordon field being located with respect to the accelerated Ψ (t1) Ψ (t2) =− csch KG KG 0A 4π 2 field. The presence of accelerated field modifies certain classical notions about space and time [8]. Thus, this approach provides a which is equivalent to the thermal-field correlation function (10) profound conceptual shift in the understanding of Unruh effect. with temperature The vacuum of an accelerated quantum field theory is filled with α T = . particles, not because the concept of the number of particles is 2π frame dependent, but because the classical notions of space and The meaning of this result is that an inertial observer, being in time are changed. the ground state of a quantum field, responds differently in the Although Unruh thermal bath is accepted by the majority of presence of an accelerated field. The effect of the accelerated field physicists, the phenomenon still lacks observational confirmation is to promote the zero-point quantum field fluctuations to the level [21]. Recently, an experiment based on classical physics was pro- of thermal fluctuations. posed to confirm the existence of Unruh effect [22]. The analysis L.C. Céleri, V.I. 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