Bohmian Field Theory on a Shape Dynamics Background and Unruh ...

Bohmian Field Theory on a Shape Dynamics Background and Unruh Effect Furkan Semih D¨ undar and Metin Arık

arXiv:1706.05890v1 [gr-qc] 19 Jun 2017

Physics Department, Bo˘gazi¸ci University, 34342, Istanbul, Turkey∗ In this paper, we investigated the role of accelerated observers in observing the Unruh radiation in the Bohmian field theory on a shape dynamics background setting. Since metric and metric momentum are real quantities, the integral kernel to invert the Lichnerowicz-York equation for first order deviations due to existence of matter terms turns out to be real. This fact makes the interaction Hamiltonian real. On the other hand, since the ground state wave functional for rectilinear observers is also real, the jump rate of the Bohmian field theory turns out to vanish. Hence we conclude that Unruh effect in Bohmian field theory on a shape dynamics background setting is non-existent. It is also found out that the non-existence of Unruh radiation is independent from the compact nature of space in shape dynamics. Therefore, observation of Unruh radiation may be a test for quantum field theory and Bohmian field theory on a shape dynamics background.

I.

INTRODUCTION

Shape dynamics (SD) [1, 2] (see ref. [23] for a review) is a theory of gravitation that is based on 3-conformal geometries where time is absent, based on the Julian Barbour’s interpretation of the Mach’s principle [3]. Although SD agrees with general relativity locally (hence passing all the tests in the low curvature regime) it may have global differences: For example the spherically symmetric vacuum solution is not the Schwarzschild black hole but rather a wormhole [16]. Moreover space-time is an emergent phenomenon is SD and it exactly ceases to emerge at the event horizon of the spherically symmetric vacuum solution of SD [16]. This fact may be related to the firewall paradox and may be the solution of firewall paradox. Bohmian mechanics (BM) [4, 13, 14] (see ref. [12] for a review) is an observer-free interpretation of quantum mechanics. There, the quantum particles have definite positions whereas the whole system is always |ψ|2 distributed. Hence the Born’s statistical interpretation of quantum mechanics is respected. On the other hand Bohmian field theory (BFT) [5, 6, 8–11, 24–26] is an attempt to give quantum field theory a Bohmian interpretation. In this paper we give a derivation for the non-existence of Unruh effect [27] from the perspective of Bohmian field theory. For this purpose we use a 3-dimensional shape dynamics theory on top of which there is a massive scalar Bohmian field. In order to quantify the non-existenece of Unruh effect, we use the jump rates (explained in section IV) from Bohmian field theory. For that purpose we summarize SD and BFT in sections II and IV. We give a description of scalar fields on shape dynamics background following [21] in section III. In section V we give details on how to put Bohmian fields on a shape dynamics background and show the non-existence of Unruh effect in this setting.

∗

[email protected]

II.

A BRIEF ACCOUNT OF SHAPE DYNAMICS

In this part we explain SD on compact spaces. Original SD is a theory on compact spaces [1, 2]. Its configuration space is the same as that of general relativity [17, 18]. In SD, the basic ontology is conformal 3-geometries. Hence the theory is symmetric under (volume preserving) conformal transformations. If the reader is interested in the case of asymptotically flat space, ref. [15] is a good source. However, in our approach we will closely follow [21]. Let Σ, N and ξ a be a compact Cauchy surface, lapse function and shift vector field. In the ADM formalism the line element of GR is written as follows (in the mostly plus signature): ds2 = (gab ξ a ξ b − N 2 )dt2 + 2gabξ a dxb dt + gab dxa dxb . (1) It is found out that N, ξ a turn out to be Lagrange multipliers of the following constraints:  π ab (g g − g g /2)π cd ac bd ab cd √ g Σ  √ − (R − 2Λ) g + matter terms , Z d3 x(π ab Lξ gab + π A Lξ φA ), H(ξ) =

S(N ) =

Z

d3 xN

(2) (3) (4)

Σ

where φA and π A stand for matter fields and their conjugate momenta. Time evolution is generated by S(N ) + H(ξ). In the constant mean extrinsic curvature √ gauge fixing of GR (π − hπi g = 0) there is a conformal factor Ω0 that solves the Lichnerowicz-York (LY) equation:  hπi2 1 8∇ Ω = RΩ + − 2Λ Ω5 − Ω−7 σ ab σab 6 g + matter terms, (5) 2



where σ ab ≡ π ab − πg ab /3 is the trace-free part of the conjugate momentum of the metric. It turns out that the

2 R √ York Hamiltonian, Σ d3 x gΩ60 , generates time evolution τ = 2hπi/3 where π = π ab gab and hπi = R time, Rin York √ g. The smeared gauge-fixging condition Σ π/ Σ   3 √ d3 xρ π − τ g Q(ρ) = 2 Σ Z

(6)

functions “as a generator of spatial conformal transformation for the metric and the trace-free part of the metric momenta and forms a closed constraint algebra with the spatial diffeomorphism generator” [21]. Therefore the gauge-fixing condition (6) can be used to map √ π → 32 τ g and from now on τ becomes an “abstract evolutionR parameter” [21] and the total volume of the space, √ V = Σ d3 x g, is a pure gauge entity. This new theory is called as shape dynamics with Hamiltonian HSD =

quad Hmatt =

, φA , π A ].

H1 = 6

Z

√ quad d3 x d3 y g(x)Ω50 (x)K(x, y)Hmatt (y).

(8)

quad where Hmatt is quadratic in field and its momenta and K(x, y) is the Green’s function given by:

Z

IV.

d3 x

√ g(x)Ω50 (x)K(x, y).

(12)

A BRIEF ACCOUNT OF BOHMIAN FIELD THEORY

BFT is expressed as a stochastic field theory [10]. Path that the system follows in the configuration space (Q) is piece-wise continuous where jumps occur when particles are emitted or absorbed [10]. Let the total Hamiltonian of the system be written as H = H0 +HI where H0 is the free Hamiltonian and HI is the interaction Hamiltonian. In the continuous part of the path in Q the system evolves according to the following formula [10]: Ψ∗ (Qt )(qˆ˙Ψt )(Qt ) dQt , =ℜ t dt |Ψt (Qt )|2

(9)

(13)

where ℜ(·) stands for the real part of the expression inside the parenthesis and d qˆ˙ = eiH0 t qˆe−iH0 t = i[H0 , qˆ]. dt

(14)

Here the wave-function Ψt (q) evolves according to the Schr¨odinger equation. When jumps occur, σ Ψ (dq|q ′ ) is the jump rate in dt seconds from a point q ′ ∈ Q to a volume dq around a point q ∈ Q [10]: σ Ψ (dq|q ′ ) = 2

h  3 K(x, y) = 8∇2 − R + 5( τ 2 − 2Λ)Ω40 8 σ ab σab −8 i−1 , Ω0 + g

(11)

Σ

Let us begin with a usual scalar field on a shape dynamics background. The steps are already sketched in [21] and we will follow it closely in this regard. We suppose two properties of the setting: 1) the perturbation theory is valid and 2) we look for description on short time intervals. First property is required to use the perturbation theory and the second one is needed to disregard back-reaction of the field on space. We assume that the Hamiltonian, HSD , is even in field quantities, and accepts a perturbative expansion HSD = H0 +εH1 +O(ε2 ). Let φ and ̟ be the massive scalar field and its conjugate field momentum and Ω = Ω0 + εΩ1 + O(ε2 ) be the solution of the LY equation (5), where Ω0 is solves the source-free LY equation. We expand the LY equation as a perturbative series in ε and the result to first order in ε is the following: H quad d yK(x, y) matt Ω1 = √ (y), Ω0 g Σ

 ̟2 √ √ + (Ω40 g ab ∂a φ∂b φ + m2 φ2 ) g . (10) g

Since the lapse function N (x) is the Lagrange multiplier of the Hamiltonian, we obtain:

(7)

3



N (y) = 6

CLASSICAL MASSIVE SCALAR FIELD ON A SHAPE DYNAMICS BACKGROUND

Z

1 2

Moreover, the first order Hamiltonian, H1 , is given as follows:

3 √ √ d3 x gΩ60 [gab , π ab → σ ab + τ g ab g 2 Σ

Z

For its relation to GR using the linking theory approach, readers may see [17, 18]. III.

where we used τ = 2hπi/3. For a massive scalar field we have:

[ℑΨ∗ (q)hq|HI |q ′ iΨ(q ′ )]+ , |Ψ(q ′ )|2

(15)

where ℑ(·) stands for the imaginary part of the expression inside the parenthesis and x+ expresses the positive part of x, i.e. max(x, 0).

3 V. MASSIVE SCALAR BOHMIAN FIELD ON A SHAPE DYNAMICS BACKGROUND AND THE UNRUH EFFECT

In BFT there is no Lorentz symmetry, hence it is more prone to quantization in SD point of view. The existence of conformal symmetry in SD is expected to add a different flavor to BFT. In our approach to the problem, we embrace a semi-classical perspective where the SD is classical and the field is quantized. For BFT, the equations of motion are given in equations (13) and (15) where the interaction Hamiltonian for the field is given in equation (11). In SD, the field equation is given by the following formulae [21]: φ˙ = {φ, H1 } + O(e),

̟ ˙ = {̟, H1 } + O(e).

(16)

First, it does not look manifestly if these two laws of evolution agree with each other or not. However, we can make them agree manifestly using the Schr¨odinger wavefunctional for the field. The wavefunctional equation of motion is as follows [22]:

i

∂Ψ[φ, t] 1 = ∂t 2

 δ2 d3 x − 2 δφ Σ

Z

 − g ab ∂a φ∂b ψ + m2 φ2 Ψ[φ, t].

2 #  Z " 2 Z 1 δG δ G = E0 − − 2 + (|∇φ|2 + m2 φ2 ). δφ δφ 2 Σ Σ (18)

Let us consider Σ = T3 , e.g. Qthe 3-torus with side lenghts L1 , L2 and L3 such that i Li = V . The |∇φ|2 is a boundary term and since Σ has no boundary it vanishes. At this point our treatment of the issue differs from [20]. The equation satisfied by G[φ] is found to be the following: δ2 G − δφ2



δG δφ

2

=−

2E0 + m2 φ2 . V

∇2x G(~x, ~y) = δ(~x − ~y) −

(19)

This is a first order variational equation in δG/δφ and it can be solved analytically.

1 , V

(20)

where the last terms is needed for consistency when the equation is integrated and Gauss’ theorem is used. In our case, the function of interest is found to be the following:

K(x, y) =

1 |~x − ~y |2 − . 8|~x − ~y | 48V

(21)

quad However, this is valid when the average value of Hmatt over space is zero. Otherwise the Laplace equation is not invertible on compact spaces. This gives us E0 = 0 in equation (18) and the equation can easily be solved by Mathematica using Bessel functions:

(17)

Hence, all in all, scalar field evolves according to the Schr¨odinger wavefunctional equation (17) instead of the canonical equation of motion given in (16), particles move in time according to (13) and (15). We can calculate the ground state functional (for more see [20]). Observe that the Hamiltonian in the Schr¨odinger wave functional equation (17) is time-independent, hence we set Ψ0 [φ] = e−iEt η exp(−G[φ]) and we have as the time-independent Schr¨odinger wave functional equation: 1 2

In our derivation, the metric gab = δab is flat and the result will be valid for all conformally flat metrics because of the Weyl symmetry of SD. As an ansatz we take π ab = 0. When this is the case, it is quite easy to solve equation (9) for K(x, y). It is proportional to the Green’s function for a Laplacian on 3-torus. However there is a subtlety in tori in terms of the Green’s function equation [19]. For example, the equation to be satisfied by the Green’s function on 3-torus with volume V is:

Z

φ

 − cJ−1/4 (mϕ2 /2) + m − 2J−3/4 (mϕ2 /2) 1  . − cJ−5/4 (mϕ2 /2) + cJ3/4 (mϕ2 /2) ϕ2     2 J1/2 (mϕ2 /2) + cJ−1/4 (mϕ2 /2) ϕ dϕ. (22)

G[φ] =



where c is a constant integration, and the additive constant of integration is absorbed in η. On the other hand, Ω0 is a solution of the LY equation (5) without the matter terms. The LY equation becomes ∇2 Ω0 = 0, and we take Ω0 = 1. In this case, quad Hmatt becomes: quad Hmatt

1 = 2

  δ2 a 2 2 − 2 + ∂ φ∂a φ + m φ , δφ

(23)

and H1 is the following: 3 H1 = 4

Z

Σ×Σ



1 |~x − ~y |2 − |~x − ~y | 6V



quad Hmatt (y).

(24)

Unruh effect [27], is the observance of radiation in the Minkowski vacuum state of rectilinear observers from the perspective of accelerated observers. In its derivation, Lorentz symmetry of space-time and the field are used. It is therefore an open question whether this effect is existent or not in Bohmian field theory on a shape dynamics background.

4 When σ Ψ in equation (15) is calculated using the values and functions obtained so far, it turns out to be zero. quad This is because the action of Hmatt on Ψ0 [φ] reproduces a real function. Hence the system stays in the ground state forever. This is a consistent result within the common wisdom, because all the quantities calculated so far in this section was for a standing-still observer which is valid for observers moving at a constant speed. From now on, we will consider a constantly accelerating observer. The three dimensional Rindler metric for the observer in the special relativity is the following: ds2 = −e2aξ dλ2 + e2aξ dξ 2 + dy 2 + dz 2 .

(25)

From the perspective of ADM formalism, the shift vector field vanishes, Lapse function equals N = eaξ and the 3-metric is the following: gab = diag(e2aξ , 1, 1).

(26)

First of all observe that this metric is not conformally flat. Hence it falls down into a different shape dynamics description, though it still describes a three torus Σ (all one needs to do is to map eaξ dξ → ±dx). However when equation (9) is solved for K(x, y) whatever the π ab (since it is real) the function K(x, y) will turn out to be real. When the interaction Hamiltonian HI is put in equation (15) with the ground state wave functional Ψ0 [φ] the results turn out to be real. Hence we have σ Ψ = 0. Therefore there is no Unruh radiation to be observed.

VI.

CONCLUSION

In this paper we investigated Bohmian field theory on a shape dynamics background. Our aim has been to

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ACKNOWLEDGEMENTS

Authors are grateful to Tim Koslowski, Mikhail Sheftel, Cihan Pazarba¸sı and Bayram Tekin for useful dis˙ cussions. F.S.D. is supported by TUBITAK 2211 Scholarship. The research of F.S.D. and M.A. is partly supported by the research grant from Bo˘ gazi¸ci University Scientific Research Fund (BAP), research project No. 11643.

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