We investigate the rigidly rotating quantum thermal distribution of fermions in flat space-time. We find that thermal states diverge on the speed of light surface. We remove the divergences by enclosing the system inside a cylindrical boundary and investigate thermal expectation values and the Casimir effect for two sets of boundary conditions

Ambruş, V.E.

Winstanley, E.

White Rose Research Online

arXiv:1302.3791v2  [gr-qc]  18 Feb 2013ROTATING FERMIONSVICTOR E. AMBRUŞ and ELIZABETH WINSTANLEYSchool of Mathematics and Statistics, University of Sheffield, United KingdomFebruary 19, 2013AbstractWe investigate the rigidly rotating quantum thermal distribution of fermions in flat space-time. We find that thermal states diverge on the speed of light surface. We remove the diver-gences by enclosing the system inside a cylindrical boundary and investigate thermal expectationvalues and the Casimir effect for two sets of boundary conditions.1 IntroductionThe authors of Ref. [4] have shown that thermal states are ill-defined for quantum scalar particles onan unbounded, rigidly rotating, space-time, but are regular if the system is enclosed in a boundaryinside the speed of light surface (SOL).We perform a similar analysis for fermions. We discover that the difference between Bose-Einstein and Fermi-Dirac statistics allows fermions to exist in thermal states which diverge on theSOL, but are perfectly regular if enclosed inside a boundary located within the SOL.2 Rotating fermions on the unbounded Minkowski space-timeWe consider a Minkowski space-time rotating with a constant angular velocity Ω. The line elementin cylindrical coordinates, with the z axis pointing along Ω, can be obtained from the Minkowskimetric by making the substitution ϕ→ ϕ+Ωt:ds2 = −(1− ρ2Ω2)dt2 + 2ρ2Ω dt dϕ+ dρ2 + ρ2dϕ2 + dz2. (1)We define the surface ρΩ = 1 as the speed of light surface (SOL), since on this surface, co-rotatingobservers move at the speed of light.The solutions of the Dirac equation on the metric in Eq. (1) can be written as:Uk(x) =12πe−iẼktuk(ρ, ϕ, z), (2)where uk is a four-spinor and k is a generic label distinguishing between independent modes. Theeigenvalue Ẽk of the Hamiltonian is related to the Minkowski energy Ek through Ẽk = Ek−Ω(m+12),where m+ 12is the eigenvalue of the angular momentum along the rotation axis.1If bk is the creation operator for quanta described by the modes in Eq. (2), we can constructthe following thermal expectation value (t.e.v.) at inverse temperature β [8]:〈b†kbk′〉β=δkk′eβẼk + 1. (3)For the vacuum expectation value of products of the form b†kbk′ to vanish, the above expression mustgo to 0 as β → ∞. Therefore, we must restrict particle modes to modes with positive frequency(i.e. positive Ẽk), as discussed in Ref. [6]. Allowing particle modes to have negative frequencywould give rise to temperature-independent terms in t.e.v.’s, similar to those obtained in Ref. [8].We now present the t.e.v. of the energy density using the quantisation proposed by Ref. [6](ε = 1− ρ2Ω2):〈: Ttt :〉β =7π260β4ε+Ω28β2(43ε2−13ε). (4)Similar expressions can be obtained for the other non-vanishing components of the stress-energytensor (i.e. for Ttϕ, Tρρ, Tϕϕ and Tzz), as well as for the neutrino current parallel to the rotationaxis [1]. A common feature of these t.e.v.’s is their divergent behaviour as the SOL is approached,which we illustrate by the thin curves in Fig. 1.For the scalar field, the indefinite norm requires particles to be described by modes of positivenorm [7], forcing negative frequency modes into the set of particle modes. Moreover, the Bose-Einstein density of states factor diverges for frequencies approaching 0, even when the correspondingMinkowski energy is finite, requiring thermal states to have an infinite energy density at each pointin the space-time.ΡWLog@TttD0.2 0.4 0.6 0.8 1.-3-2-1012ΒW=2.ΒW=1.25ΒW=1.ΒW=0.8Unbounded(a)ΡWLog@TttD0.2 0.4 0.6 0.8 1.-2-1012345ΒW=2.ΒW=1.25ΒW=1.ΒW=0.8Unbounded(b)Figure 1: The t.e.v. of the energy density in the (a) spectral and (b) MIT bag models for thermalfermions inside a boundary of radius R = Ω−1 is compared with the corresponding energy densityfor the unbounded space-time (thin curves), for four values of the inverse temperature β (β isincreasing from top to bottom). The two models give different results near the boundary: in thespectral case, Ttt decreases to a local minimum, then increases to a finite value on the SOL, whilein the MIT bag model, Ttt increases to a local maximum, then decreases to a finite value on theSOL which is more than two times greater than in the spectral case.23 Rotating fermions inside an infinite cylinderThe divergent nature of expectation values in the rigidly rotating space-time is related to theinclusion of a region of space which co-rotates at speeds larger than the speed of light. FollowingRef. [4], we eliminate this region by enclosing the system inside a cylindrical boundary of radius R,centred on the rotation axis, such that RΩ ≤ 1. We use two sets of boundary conditions, describedbelow.For consistency, a set of boundary conditions must ensure the vanishing of the time derivativeof the Dirac inner product, given by:i∂t 〈ψ,χ〉 =∫∂VdΣψnµγµχ, (5)where n = nµdxµ is the normal to the boundary. In the spectral model [5], Eq. (5) is satisfied modeby mode by Fourier coefficients of the wave functions ψ and χ. In order to preserve the chargeconjugation invariance of the theory, the boundary conditions in the spectral model depend on thespectral index of the mode, making this formulation non-local. By contrast, the MIT bag model [2]satisfies Eq. (5) in a purely local way, by requiring that the wave functions satisfy the equationinµγµψ = ψ. Plots of the t.e.v. of the energy density in the two models are presented in Fig. 1.4 The Casimir divergenceThe presence of a boundary in a quantum system can induce a change in its vacuum state. Thedifference between the expectation value of the energy density for fermions of mass µ in the vacuumstates of the bounded and unbounded systems is found to diverge as an inverse power of the distanceto the boundary:〈Ttt〉Casimir ∼−1120π2R4×1 + 10µR(1− ρ/R)3(MIT bag model)−116π2R4×1(1− ρ/R)4(spectral boundary conditions.)(6)According to Ref. [3], the energy density should diverge as the inverse cube of the distance tothe boundary for a purely local stress-energy tensor. Although it seems to violate this result, thespectral model gives a different order for the divergence because it does not satisfy the assumptionof locality.AcknowledgmentsThis work is supported by the Lancaster-Manchester-Sheffield Consortium for Fundamental Physicsunder STFC grant ST/J000418/1, the School of Mathematics and Statistics at the University ofSheffield and European Cooperation in Science and Technology (COST) action MP0905 “BlackHoles in a Violent Universe”.References[1] Victor E. Ambruş and Elizabeth Winstanley. Rotating fermions. paper in preparation.3[2] A. Chodos, R.L. Jaffe, K. Johnson, C.B. Thorn, and V.F. Weisskopf. New extended model ofhadrons. Phys. Rev. D, 9(12):3471–3495, 1974.[3] D. Deutsch and P. Candelas. Boundary effects in quantum field theory. Phys. Rev. D,20(12):3063–3080, 1979.[4] G. Duffy and A.C. Ottewill. Rotating quantum thermal distribution. Phys. Rev. D, 67(044002),2003.[5] M. Hortacşu, K.D. Rothe, and B. Schroer. Zero-energy eigenstates for the Dirac boundaryproblem. Nucl. Phys., B171:530–542, 1980.[6] B. Iyer. Dirac field theory in rotating coordinates. Phys. Rev. D, 26(8):1900–1905, 1982.[7] J.R. Letaw and J.D. Pfautsch. Quantized scalar field in rotating coordinates. Phys. Rev. D,22(6):1345–1351, 1980.[8] A. Vilenkin. Quantum field theory at finite temperature in a rotating system. Phys. Rev. D,21(8):2260–2269, 1980.4

Rotating fermions

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Rotating fermions

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