New Geometric Transition as Origin of Particle Production in Time-Dependent Backgrounds
aa r X i v : . [ h e p - t h ] A ug YITP-13-52
New Geometric Transition as Origin of Particle Production in Time-DependentBackgrounds
Sang Pyo Kim ∗ Department of Physics, Kunsan National University, Kunsan 573-701, Korea † andYukawa Institute for Theoretical Physics, Kyoto University, Kyoto 606-8502, Japan (Dated: October 10, 2018)By extending the quantum evolution of a scalar field in time-dependent backgrounds to thecomplex-time plane and transporting the in-vacuum along a closed path, we argue that the geometrictransition from the simple pole at infinity determines the multi-pair production depending on thewinding number. We apply the geometric transition to Schwinger mechanism in the time-dependentvector potential for a constant electric field and to Gibbons-Hawking particle production in theplanar coordinates of a de Sitter space. PACS numbers: 12.20.-m, 04.62.+v, 03.65.Vf
The interaction of quantum field with a background gauge field or spacetime can produce particles as a nonpertur-bative quantum effect. Schwinger mechanism is pair production by a constant electric field, which provides enoughenergy to separate charged pairs from the Dirac sea [1]. Hawking radiation is the emission of particles from vacuumfluctuations, which are separated by the horizon of a black hole [2]. Recently Hawking radiation has been interpretedas quantum tunneling of virtual pairs near the horizon of the black hole [3]. Also it has been known for long that anexpanding spacetime produces particles [4] and de Sitter (dS) radiation has a thermal distribution [5].In quantum field theory the out-vacuum of a quantum field may differ from the in-vacuum through the interactionwith a background field and may be expressed as multi-particle states of the in-vacuum via the Bogoliubov transfor-mation [6]. In the in-out formalism the scattering matrix between the in-vacuum and the out-vacuum determines theprobability for the in-vacuum to remain in the out-vacuum, which in turn is given by the pair-production rate forbosons or fermions [7]. The exact vacuum persistence and pair production requires the solution of the quantum fieldin the background field. With limited knowledge of exact solutions, approximation scheme proves a practical approachor provides an intuitive understanding for pair production. Various approximation schemes have been proposed forHawking radiation [8].Each Fourier mode of a scalar field in the time-dependent vector potential for a constant electric field and in theplanar coordinates of a dS space describes the scattering problem over a potential barrier. In the phase-integralmethod Schwinger pair-production rate is determined by the action integral among quasi-classical turning points inthe complex plane of time or space [9, 10]. In particular, the action integral has been proposed as the contour integralin the complex plane for Schwinger mechanism [9] and for Hawking radiation [11]. Furthermore, the Stokes lines andanti-Stokes lines for more than one pair of quasi-classical turning points distinguish boson and fermion pair production[10] and the dimensionality of particle production in dS spaces [12]. The complex analysis has been used in connectionwith particle production [13, 14], the instanton action [15, 16], and the worldline instanton [17, 18].In this paper we propose the geometric transition of the Hamiltonian in the complex-time plane as a new interpreta-tion of particle production in time-dependent backgrounds. It has been observed that a time-dependent Hamiltoniancan have the geometric transition in the complex-time plane [19–21]. In the functional Schr¨odinger picture eachFourier mode of quantum field in a time-dependent background has the Hamiltonian with time-dependent frequencyand/or mass. We argue that the evolution of the in-vacuum along a complex closed path of non-zero winding numberleads to the geometric transition from the simple pole at infinity and results in particle production, in strong con-trast with the trivial real-time evolution without level-crossings. The evolution of the in-vacuum along a path in thecomplex-time plane is reminiscent of the closed-time path integral in the in-in formalism [7].A complex scalar field with mass m and charge q in a constant electric field in the (d+1)-dimensional Minkowskispacetime has the Fourier-decomposed, time-dependent Hamiltonian [in units of c = ~ = 1] H ( t ) = Z d d k h π † k π k + 12 ω k φ † k φ k i , (1) † Permanent address ∗ Electronic address: [email protected] where ω k ( t ) = m + k ⊥ + ( k k + qEt ) . (2)Here k ⊥ and k k are the transverse and longitudinal momenta and the vector potential is A k ( t ) = − Et , which providesa time-dependent background. The complex scalar field is equivalent to two real scalar fields: one for particle and theother for antiparticle. In the planar coordinates of the (d+1)-dimensional dS space ds = − dt + e H HC t d x d , (3)where H HC is the Hubble constant, a massive real scalar field has the time-dependent Hamiltonian H ( t ) = Z d d k h M ( t ) π k + M ( t )2 ω k ( t ) φ k i , (4)where M ( t ) = e dH HC t , ω k ( t ) = m + k e H HC t . (5)In the functional Schr¨odinger picture, the quantum state obeys the time-dependent Schr¨odinger equation i ∂∂t | Ψ( t ) i = ˆ H ( t ) | Ψ( t ) i , | Ψ( t ) i = Y k | Ψ k ( t ) i . (6)For the purpose of this paper, it is sufficient to consider the time-dependent oscillator defined along the real-timeaxis as H ( t ) = 12 M ( t ) p + M ( t )2 ω ( t ) q , ( ω ( t ) > . (7)In the real-time evolution, the annihilation and creation operatorsˆ a ( t ) = r M ( t ) ω ( t )2 ˆ q + i p M ( t ) ω ( t ) ˆ p, ˆ a † ( t ) = r M ( t ) ω ( t )2 ˆ q − i p M ( t ) ω ( t ) ˆ p (8)diagonalize the Hamiltonian as ˆ H ( t ) = ω ( t ) (cid:16) ˆ a † ( t )ˆ a ( t ) + 12 (cid:17) . (9)Thus, an initial state | Ψ( t ) i = ˆ U ( t, t ) | Ψ( t ) i , (10)evolves by the evolution operator, which can be given by the time-ordered integral or product integral [22]ˆ U ( t, t ) = T exp (cid:2) − i Z tt ˆ H ( t ′ ) dt ′ (cid:3) = t Y t exp (cid:2) − i ˆ H ( t ′ ) dt ′ (cid:3) . (11)In terms of the number states (9), the evolution operator can be further written as [21]ˆ U ( t, t ) = Φ T ( t )T exp (cid:2) − i Z tt (cid:0) H D ( t ′ ) − A T ( t ′ ) (cid:1) dt ′ (cid:3) Φ ∗ ( t ) , (12)where H D ( t ) and Φ( t ) denote the diagonal matrix and the column vector, respectively, H D ( t ) = ω ( t ) . . . n + . . . , Φ( t ) = | , t i ... | n, t i ... , (13)and A ( t ) is the induced vector potential A ( t ) = i Φ ∗ ( t ) ∂ Φ T ( t ) ∂t = i ˙ ω ( t )4 ω ( t ) (cid:0)p n ( n − δ mn − − p ( n + 1)( n + 2) δ mn +2 (cid:1) . (14)Here and hereafter overdots denote derivatives with respect to the real or complex time. Hence ˆ U ( t , t ) from t to any future time t and back to t along the real-time axis becomes unity since H D ( t ) and A ( t ) do not have anysingularity due to ω ( t ) >
0, which is the case of charged scalars in a time-dependent vector potential or real scalarsin a dS space. Note that the path from t to t along the real-time axis is a loop of zero-winding number, which willbe denoted as C (0) ( t ) with the base point t . In other words, ˆ U ( C (0) ( t )) = I in the complex-time plane and thescattering amplitude between the in-vacuum and the transported in-vacuum is unity along the real-time axis h , C (0) ( t ) | , t i = 1 . (15)The in-in formalism thus becomes trivial as long as the real time is concerned.However, a time-dependent Hamiltonian, provided that it has a level-crossing in the complex-time plane, leads tothe geometric transition amplitude, which is responsible for an exponential decay of the initial state and the transitionto other states [19, 20]. In a similar manner, let us extend the Hamiltonian H ( t ) to the complex-time plane and assumethat H ( z ) is analytic and the orthonormality h m, z | n, z i = δ mn holds. In a properly chosen Riemann sheet in thecomplex-time plane, the frequencies (2) and (5) have two branch points of the form ω ( z ) = f ( z ) q ( z − z )( z − z ∗ ) , (16)where f ( z ) is an analytic function. Though the complex frequency (16) has level-crossings in the whole complex plane,we cut two branch lines from z and z ∗ as shown in Fig. 1, which make ω ( z ) analytic in the proper Riemann sheet.Defining ˜ H ( z ) := H D ( z ) − A T ( z ), we notice that the matrix-valued ˜ H ( z ) does not commute with ˜ H ( z ′ ) in generalfor z = z ′ unless ω ( z ′ ) ˙ ω ( z ) = ω ( z ) ˙ ω ( z ′ ). Then, without level-crossings, all the time-ordered integrals of ˆ H ( z ) in thecomplex-time plane along closed paths of the same winding number and with the same base point t on the real-timeaxis are equal to each other [22]T exp (cid:2) − i I C I ( t ) ˜ H ( z ) dz (cid:3) = T exp (cid:2) − i I C II ( t ) ˜ H ( z ) dz (cid:3) . (17)Hence the time integral in the complex-time plane is independent of paths and depends only on the homotopy class ofwinding numbers. In fact, the path in the left panel of Fig. 1 has the winding number 1 and is equivalent to anotherpath C (0) ( t ) along the real-time axis plus the loop C (1) (0) of the winding number 1 encircling z = 0 while the path C (3) ( t ) in Fig. 2 has the winding number 3. Thus, since the time integral along C (0) ( t ) in the real-time axis is unity,the time integral along a curve C ( t ) consisting of C (0) ( t ) and C ( n ) (0) of winding number n around z = 0 becomesT exp (cid:2) − i I C ( t ) ˜ H ( z ) dz (cid:3) = T exp (cid:2) − i I C ( n ) (0) ˜ H ( z ) dz (cid:3) (18)However, in the lowest order of the Magnus expansion [23], the residue theorem holds for the matrix-valued ˜ H ( z )[22] T exp (cid:2) − i I C ( n ) (0) ˜ H ( z ) dz (cid:3) = exp (cid:2) − πn Res[ ˜ H ( z )] (cid:3) , (19)where Res[ ˜ H ( z )] is the residue of the simple pole z at infinity [24]. In fact, the frequencies (2) and (5) do havethe simple pole at infinity, so the scattering amplitude between the in-vacuum and the transported in-vacuum isapproximately given by h , C ( n ) ( t ) | , t i = e − πn Res[ ω ( z = ∞ )] , (20)where the factor of 1 / n is the winding number. The dynamicalphase does not contribute to the scattering amplitude since it returns to t and the frequency does not have any finitesimple poles. The exponentially decaying scattering amplitude implies transitions to excited states, that is, particleproduction. The residue at infinity is found by the large z -expansion of the complex frequency (16) ω ( z ) = f ( z ) h z − z + z ∗ − ( z − z ∗ ) z + · · · i . (21) FIG. 1: The frequency ω ( z ) has two branch points at Z and Z ∗ , which are isolated by two branch cuts, and has the simplepole located at Z = ∞ . The path C (1) ( t ) starts from the base point t , follows a loop clockwise and returns to t , excludingthe simple pole at the infinity [left panel]. The equivalent path consists of a real-line segment C (0) ( t ) from t to t and a loop C (1) (0) encircling z = 0 [right panel].FIG. 2: The path C (3) ( t ) starts from the base point t , follows clockwise a loop of winding number 3 and returns to t . In the first case of charged scalars in the constant field, the proper Riemann sheet is the entire complex plane withbranch cuts as shown in Fig. 1. Then the magnitude square of the scattering amplitude |h , C ( n ) ( t ) | , t i | = e − nπ m k ⊥ qE , (22)is the Schwinger pair-production rate for n -pairs of charged particles and antiparticles. It is analogous to the multi-instanton actions for pair production in the Coulomg gauge for static electric fields [15]. In the second case of realscalars in the dS space, the proper Riemann sheet − π/H HC < Im t ≤ π/H HC and a conformal mapping e H HC t = z may be chosen, in which the scattering amplitude square |h , C ( n ) ( t ) | , t i | = e − nπ mHHC , (23)is the Boltzmann factor for Gibbons-Hawking radiation in the dS space.In summary, we showed that the geometric transition from a simple pole at infinity in the complex-time planecould explain particle production in a constant electric field and in a dS space. The Fourier-decomposed Hamiltonianfor charged scalars in the constant electric field and for real scalars in dS space is infinite number of oscillators withtime-dependent frequencies and/or mass. In the real-time evolution, any state prepared at an initial time that evolvesinto a future time and returns to the initial time remains in the same state with a trivial phase factor since there is nolevel-crossing, so the scattering amplitude between the in-vacuum and the transported in-vacuum is unity. However,we argued that the evolution along a closed path in the complex-time plane obtains a new geometric transitioncoming from the residue of the simple pole at the infinity, which differs from the geometric transition coming fromlevel-crossings. Further, the scattering amplitude between the in-vacuum and the in-vacuum transported along a pathof winding number n leads to production of n -pairs.Finally, a few comments are in order. First, it is worth to note that the geometric transition for the in-vacuum,though a consequence of nonstationarity of the Hamiltonian, resolves the factor of two puzzle for tunneling interpre-tation of Hawking radiation. The factor of two puzzle was explained in different ways [25–28] and by including thetemporal contribution under the coordinate transformation from the embedding geometry [29–31]. The scatteringamplitude between the in-vacuum and the transported in-vacuum takes the vacuum energy into account, which countsonly a half of the energy quanta. Second, the massless limit of eq. (23) gives a unity scattering amplitude square be-tween the in-vacuum and the transported in-vacuum in the complex plane as in eq. (15). In fact, under the conformalmapping z = e H HC t the infinity has a double pole and thus does not contribute to the residue, which implies thatthe probability for the transported in-vacuum to remain in the in-vacuum is unity. The result is consistent with noproduction of massless particles in dS spaces in the in-out formalism, but there are subtle issues in the massless limit[32, 33]. Third, the geometric transition can be generalized to a frequency that has many level crossings and finitesimple poles, which is the case of generic time-dependent vector potentials and global coordinates of dS spaces andthe Friedmann-Robertson-Walker spacetime [34]. Then the homotopy classes of paths are classified by finite simplepoles inside the loops. The homotopy classes may have something to do with the Stokes phenomenon for particleproduction [10, 12], which is beyond the scope of this paper. Another issue not pursued in this paper is the stimulatedpair production from an initial particle state. Acknowledgments
The author thanks Eunju Kang for drawing figures. This paper was initiated during the Asia Pacific School onGravitation and Cosmology (APCTP-NCTS-YITP Joint Program) in Jeju, Korea in 2013, benefited from helpfuldiscussions at the Workshop on Gravitation and Numerical Relativity (APCTP Topical Program) and completed atYukawa Institute for Theoretical Physics, Kyoto University. This work was supported by Basic Science ResearchProgram through the National Research Foundation of Korea (NRF) funded by the Ministry of Education (NRF-2012R1A1B3002852). [1] J. Schwinger, Phys. Rev. 82 (1951) 664.[2] S.W. Hawking, Commun. Math. Phys. 43 (1975) 199.[3] M.K. Parikh, F. Wilczek, Phys. Rev. Lett. 85 (2000) 5042.[4] L. Parker, Phys. Rev. 183 (1969) 1057.[5] G.W. Gibbons and S.W. Hawking, Phys. Rev. D 15 (1977) 2738.[6] B.S. DeWitt, Phys. Rep. 19 (1975) 295.[7] B.S. DeWitt, The Global Approach to Quantum Field Theory, Oxford University Press, New York, U.S.A., 2003.[8] L.Vanzo, G. Acquaviva, R. Di Criscienzo, Class. Quantum Grav. 28 (2011) 183001.[9] S.P. Kim, D.N. Page, Phys. Rev. D 75 (2007) 045013.[10] C.K. Dumlu, G.V. Dunne, Phys. Rev. Lett. 104 (2010) 250402.[11] S.P. Kim, JHEP 11 (2007) 048.[12] S.P. Kim, JHEP 09 (2010) 054.[13] E. Keski-Vakkuri, P. Kraus, Phys. Rev. D 54 (1996) 7407.[14] K. Srinivasan, T. Padmanabhan, Phys. Rev. D 60 (1999) 024007.[15] S.P. Kim, D.N. Page, Phys. Rev. D 65 (2002) 105002.[16] S.P. Kim, D.N. Page, Phys. Rev. D 73 (2006) 065020.[17] G.V. Dunne, C. Schubert, Phys. Rev. D 72 (2005) 105004.[18] G.V. Dunne, Q-h. Wang, H. Gies, C. Schubert, Phys. Rev. D 73 (2006) 065028.[19] J.-T. Hwang, P. Pechukas, J. Chem. Phys. 67 (1977) 4640.[20] A. Joye, H. Kunz, Ch.-Ed Pfister, Ann. Phys. 208 (1991) 299.[21] S.P. Kim, “Matrix Operator Approach to Quantum Evolution Operator and Geometric Phase,” SNUTP-92-10[arXiv:1212.2680].[22] J.D. Dollard, C. N. Friedman, Product Integral, Addison-Wesley, Massachusetts, U.S.A., 1979.[1] J. Schwinger, Phys. Rev. 82 (1951) 664.[2] S.W. Hawking, Commun. Math. Phys. 43 (1975) 199.[3] M.K. Parikh, F. Wilczek, Phys. Rev. Lett. 85 (2000) 5042.[4] L. Parker, Phys. Rev. 183 (1969) 1057.[5] G.W. Gibbons and S.W. Hawking, Phys. Rev. D 15 (1977) 2738.[6] B.S. DeWitt, Phys. Rep. 19 (1975) 295.[7] B.S. DeWitt, The Global Approach to Quantum Field Theory, Oxford University Press, New York, U.S.A., 2003.[8] L.Vanzo, G. Acquaviva, R. Di Criscienzo, Class. Quantum Grav. 28 (2011) 183001.[9] S.P. Kim, D.N. Page, Phys. Rev. D 75 (2007) 045013.[10] C.K. Dumlu, G.V. Dunne, Phys. Rev. Lett. 104 (2010) 250402.[11] S.P. Kim, JHEP 11 (2007) 048.[12] S.P. Kim, JHEP 09 (2010) 054.[13] E. Keski-Vakkuri, P. Kraus, Phys. Rev. D 54 (1996) 7407.[14] K. Srinivasan, T. Padmanabhan, Phys. Rev. D 60 (1999) 024007.[15] S.P. Kim, D.N. Page, Phys. Rev. D 65 (2002) 105002.[16] S.P. Kim, D.N. Page, Phys. Rev. D 73 (2006) 065020.[17] G.V. Dunne, C. Schubert, Phys. Rev. D 72 (2005) 105004.[18] G.V. Dunne, Q-h. Wang, H. Gies, C. Schubert, Phys. Rev. D 73 (2006) 065028.[19] J.-T. Hwang, P. Pechukas, J. Chem. Phys. 67 (1977) 4640.[20] A. Joye, H. Kunz, Ch.-Ed Pfister, Ann. Phys. 208 (1991) 299.[21] S.P. Kim, “Matrix Operator Approach to Quantum Evolution Operator and Geometric Phase,” SNUTP-92-10[arXiv:1212.2680].[22] J.D. Dollard, C. N. Friedman, Product Integral, Addison-Wesley, Massachusetts, U.S.A., 1979.