T H E K M S - C O N D I T I O N

Transcription

1 T H E K M S - C O N D I T I O N martin pauly January 3, 207 Abstract This report develops the relation between periodicity in imaginary time and finite temperature for quantum field theories, given by the KMS condition. First, the KMS condition is derived in a general setting and illustrated using the example of a complex scalar field. Second, the KMS condtion is set in relation to two general quantum field theoretic concepts, time-ordering and Wick s theorem at finite temperature. Third, the connection between horizons for accelerating observers and a temperature, that these observers measure, is outlined. Exemplary Unruh, Hawking and de Sitter temperature are derived. contents The KMS condtion 2 Free scalar field 3 2. Zero temperature Thermal states 5 3 Field theory 8 3. Time ordering Wick s theorem 9 4 Horizons and temperature 9 4. Unruh temperature Hawking temperature 4.3 De Sitter temperature 2 References 2 introduction This report describes the relation between periodicity in an imaginary time coordinate and temperature. It is structured as following: In a first section the KMS condition in its most common form is derived in general. In section two an example - the complex scalar field - is considered in order to provide a more detailed understanding of the origin of the KMS condition. Section three deals with two important field theoretic concepts and their relation to finite temperature and complex time, namely time ordering and Wick s theorem. Finally section four applies the KMS condition to the case of observer-dependent horizons. Exemplary, the Unruh, Hawking and de Sitter temperatures are calculated. In the first two sections this report follows ref [4]. the kms condtion We consider a quantum mechanical system with time-independent Hamiltonian H. An observable A is represented in the Heisenberg picture by A t = e ith Ae ith.

2 In analogy to classical statistical mechanics we define the temperature average of an observable A at temperature T = by A = e Z Tr H A, 2 setting k B = c = h = throughout this paper. Here the partition function Z is defined as Z = Tr e H, 3 where the trace in both cases runs over a basis of states of the underlying Hilbert space. In the limit of = T this definition recovers the standard vacuum expectation value familiar from ordinary quantum field theory. Note that this expression is only well-defined if all the traces are finite. For a compact manifold this is the case. In section 2. we thus focus on the case of a field in a box. Let us now define the function G + t, A, B = A tb = e Z Tr H e ith Ae ith B 4 = Z Tr e H Ae ith Be ith = AB t, where we used the cylicity of the trace as well as that two functions of the Hamiltonian commute. In analogy we define G t, A, B = BA t = B t A. 5 Note that the difference of these two functions is just the commutator G + t, A, B G t, A, B = [A t, B]. 6 Extending G + and G to imaginary times z = t + is with s, t R yields the two functions G + z, A, B = e Z Tr iz+ih Ae izh B G z, A, B = Be Z Tr izh Ae iz ih 7, which fulfill the condition G + z i, A, B = G z, A, B 8 or in terms of expectation values with z = t real A t i B = BA t. 9 This is the KMS condition, named after Kubo, Martin and Schwinger [6, 7], expressing a periodicity of Green s functions in imaginary time. This periodicity becomes more obvious if one defines a master function G by analytic continuation. We define { G G z, A, B = z, A, B if 0 < s < G + z, A, B if < s < 0 0 for the strip s, and everywhere else by the periodicity property 8. This function is holomorphic on strips separated by the cuts s = N, N Z. If now x y and the commutator and thus the difference between G + and G vanishes for small times - as is normally the case for observables at spacelike distances - these strips are connected. We obtain a unique holomorphic function G, that is defined on a connected region. This function now is -periodic in imaginary time. We are going to explore this construction in more detail in the following section using the example of a complex scalar field. 2

3 2 free scalar field In this section we study the example of a free complex scalar field. For comparison we first study the zero temperature case and then repeat the study in the finite temperature case, highlighting the differences that emerge due to finite temperature. 2. Zero temperature Consider a complex scalar field φ on the manifold R M, where M is equipped with the metric ds 2 = γ jk dx j dx k. We are restricting ourselves to the situation of a static spacetime. Note that if the time-time entry of a given metric is not one we can always make it such by a conformal transformation. The field φ is assumed to fulfill the equation of motion where K is the operator 2 t 2 + K φ = 0 2 K = γ jk [ iax] j [ iax] k + Vx 3 with, the covariant derivative with respect to the metric γ jk, and realvalued smooth functions Ax, Vx. We assume that K is self-adjoint, and that all eigenvalues of K are positive. To avoid technical complications we further assume that M is compact, such that K has discrete eigenvalues ω 2, Kψ = ω 2 ψ 4 with {ψ } the set of potentially complex normalized spatial eigenfunctions of K. One decomposes φ, φ as φt, x = [ ] ψ x e iωt a + e iωt b, 5 2ω with the creation and annihilation operators b, a. The field φ is decomposed accordingly. The annihilation and creation operators satisfy the commutation relations [ ] [ ] a, a µ = b, b µ = δ µ [ ] 6 [a, a µ ] = a, a µ = 0 and accordingly for the commutators involving b, b. Using the above expansion to compute the commutator of the fields yields [ ] [φt 2, x, φt, y] = φ t 2, x, φ t, y = 0 [ ] [ ] φt 2, x, φ t, y = φt, x, φ 0, y = ψ xψ 7 y e iωt e iω t 2ω with t = t 2 t. Additionally if t 2, x and t, y are space-like separated it is possible to change into a system where t 2 = t. In this system t = t 2 t = 0, thus the commutator vanishes for space-like distances. 3

4 s 2 t - -2 Figure : Structure of Gt, x, y in the complex plane at zero temperature, z = t + is. The window between the two branch-cuts on the real axis arises as the commutator of the two fields vanishes for space-like distances between t, x and 0, y assuming x y. Now consider the two functions G + t, x, y = 0 φt, xφ 0, y 0 = ψ xψ y e iω t 2ω G t, x, y = 0 φ 0, yφt, x 0 = ψ xψ y e iω t 2ω 8 where the superscript marks the zero temperature expectation value. Note that the difference between these functions is the commutator [ ] [ ] G + t, x, y G t, x, y = 0 φt, x, φ 0, y 0 = φt, x, φ 0, y 9 as seen before. We now introduce the complex time variable z = t + is and try to analytically continue the functions G ±. G converges in the upper half-plane, whereas G + converges in the lower half-plane due to the different signs in the exponential. The difference between these two functions along the real axis is given by the commutator. But for fixed x and y, x y, the commutator vanishes for sufficiently small times as the distance becomes space-like. Thus there is a region around t = 0 where G + and G match continuously. This can be seen in figure. Accordingly if we now want to define a maximally extended analytic continuation, there is exactly one holomorphic function G z, x, y that is defined on a connected region given by the lower half-plane, the upper half-plane and the connecting piece in the center. We thus define G G z, x, y if 0 < Imz z, x, y = G + z, x, y if 0 > Imz 20 G ± z, x, y if 0 = Imz and Rez < dx, y where dx, y is the distance of x and y with respect to the metric γ. 4

5 Focusing on the imaginary axis we define G i s, x, y = G is, x, y = ψ xψ y e ω s 2ω = dk 0 2π k ψ xe ik 0s 2 ψ ye ik 0s w2 where we assumed that s = s 2 s. The last identity follows by performing the integral over k 0 by means of the residue theorem. The final expression in 2 shows that G i s, x, y is the Green s function for the operator of the equation of motion in complex time 2 s 2 + K φ = 0, 22 meaning that it is the unique function decaying as s and solving 2 s 2 + K G i s, x, y = δ Ds 2 s δ D x y. 23 γy Thus we could also solve the Euclidean equation of motion and continue the solution analytically to obtain a solution of the original equation of motion Thermal states Now we carry out the same procedure as above for thermal two point functions. As we chose M to be compact the partition sum Z is well defined. We start by analogously defining G + φt, t, x, y = xφ 0, y = Z Tr e H φt, xφ 0, y G φ t, x, y = 0, yφt, x = Z Tr e H φ 0, yφt, x, where now the expectation values are thermal expectation values. For our field in a box the Hamiltonian is simply given by 24 H = n + n ω 25 so that the partition sum can be computed as Z = exp n + n ω n, n,n 2, =0 = 2 exp n ω = 26 2 e ω. n =0 Similarly one computes a a = Z = n, n,n 2, =0 exp e ω n =0 = e ω = e ω e ω a a = e ω ω µ n e n ω n=0 n µ + n µ ω µ n e nω 27 5

6 s 2 t - -2 Figure 2: Structure of Gz, x, y in the complex plane at finite temperature T =, z = t + is. Different strips are just copies of each other and the whole function is periodic. Branch-cuts appear at N, N Z. and all other expectation values of two a, a operators vanish, as each of the states in the Fock basis is an eigenstate of the Hamiltonian and states with different particle numbers are orthogonal. The first expression is just the expectation value of the number operator and we recover Bose-Einsteinstatistics, as expected. With these we are able to compute G ± t, x, y = ψ xψ y 2ω e ω e iωt + e ±iωt e ω. 28 Note the additional factor of e ω in the second part of the bracket. It comes from the first expectation value in 27 and will be crucial to establish the connection between imaginary time and temperature. If we now allow for complex times we see that the periodicity condition G + z i, x, y = G z, x, y 29 is fulfilled. Now to define a holomorphic function G that extends as far as possible we note that the exponential factors in the expression for G + decay exponentially if < Im z < 0 whereas for G they decay for 0 < Im z <. Again the difference between G + and G is the commutator and thus vanishes for small real times if x y. But as the function in the 0, strip is the same as the one in the, 0 strip by the periodicity condition 29 one can now use this functional equation to uniquely define the analytic continuation in both imaginary directions, yielding the structure depicted in figure 2. We thus obtain a function Gz, x, y as the analytic continuation of z, x, y with the periodicity property G ± G i We now define the function s, x, y = Gis, x, y = Gz + in, x, y = Gz, x, y for N Z. 30 ψ xψ y 2ω e ω e ω s + e ω s for 0 < s < 3 6

7 along the imaginary time axis. Using the periodicity of G i Fourier expand with coefficients in s we can c n = = 0 2πn ds e ωs 2ω e ω + e ω s e 2πins 2 + ω 2 32 yielding G i s, x, y = n= ψ xψ ye 2πins ω 2 + 2πn The frequencies 2πn of this Fourier expansion are normally called Matsubara frequencies and play a central role in condensed matter physics. Expression 33 is a Green s function for the operator defining the Euclidean equation of motion 22. However G i does not vanish for s going to infinity but instead is periodic. We thus have found the Green s function for the Euclidean equation of motion on a cylinder S M with circumference. In this form it becomes obvious how periodicity and temperature are related: a finite temperature directly gives a periodicity in imaginary time. While this result might look surprising at first it is not unexpected: essentially the connection between temperature and time arises because both are implemented via a state weighting factor, either the Boltzmann-factor or the time evolution. These just differ by a factor of i and thus time evolution in Euclidean time is closely related to the thermal weighting of states. The periodicity arises because of the trace in the thermal expectation value. The trace projects equal states onto each other, and thus an initial state n will have a high contribution to the thermal average if after being evolved with e H it is similar to n again. In the above we only considered compact manifolds. In a generalization to the case of an infinite manifold one has to replace the discrete sums above by integrals. All arguments from above still hold and allow to define a thermal Green s function and a thermal equilibrium state. However this state is not necessarily a state that can be represented by a density matrix in Fock space anymore. Problems may arise if the spectrum of the operator K extends to zero. In this case the integrals in 28 and 3 may be infrared divergent. Take as an example the mass-less scalar field on M = R d with m = 0, K = 2. For this the zero temperature Green s function is given by G i s, x, y = 2π d+ d d+ eik x y k k 2 with s = x 0 y 0 34 where the d + k integrations include an integration over k 0 conjugate to the time coordinate and over k i conjugate to the d spatial dimensions remember that M is just the spatial hypersurface. Due to a divergence in the IR no Green s functions exists for d = and accordingly in two-dimensional Minkowski spacetime. In the thermal case G i G i s, x, y = 2π d takes the form j= e 2πijs d d eik x y k. 35 k 2 + 2πj Note that now the integration only is over the d spacetime dimensions due to the periodicity in imaginary time. As d d k k d dk this diverges for d =, 2 and thus in two and three dimensional Minkowski space, respectively. Accordingly a thermal Green s function does not exist in two and three dimensional Minkowski space. This is one case of a more general statement: 7

8 The IR behavior of the thermal theory in d dimensions corresponds to the IR behavior of the theory at zero temperature in d dimension. For the case of Fermions the anti-commutator vanishes for sufficiently small times. One then defines the master function Gz, A, B = G z, A, B in the strip 0,. This introduces an additional minus sign in 30 and renders G anti-periodic and thus periodic with period 2. The above results then follow analogously. To relate this to horizons in hyperbolic spaces one analytically continues the time variable. A periodicity in this imaginary time variable arises. This periodicity is then identified with a temperature. For the Unruh and the Hawking case this provides a relatively easy and intuitive derivation of the associated temperatures. 3 field theory In this section we are going to highlight the relation of finite temperature to two major concepts of quantum field theory, namely time ordering and Wick s theorem. 3. Time ordering So far we have dealt with different Green s functions but not spoken about time-ordering. However due to the exceptional role of the Feynman propagator in quantum field theory this section goes into more detail on the role of time-ordering. In the case of the scalar field at zero temperature considered above we define the time-ordered Feynman propagator as G F t, x, y = 0 T [φt 2, xφt, x] 0 = { G + t, x, y if t 2 > t G t, x, y if t 2 < t 36 where t = t 2 t. Now G F can be obtained from our results in imaginary time if we perform a counter-clockwise rigid rotation of G i in the complex plane. From the point of view of imaginary time, the Feynman propagator in that sense is not unique. Instead it is just one prescription on how to perform the continuation from imaginary to real time. Others are equally valid but not necessarily equally useful. We can easily extend this to the thermal case by defining the Feynman propagator as G F t, x, y = G i it, x, y 37 under the prescription that the real axis is approached from above if t < 0 and from below if t > 0. This allows to write ] G F t, x, y = G [φt, t, x, y + x, φ 0, y 38 as the difference over the branch cut is just the commutator. We have seen that due to different convergence properties ordering of operators in imaginary time is crucial to the definition of the master function G. One could thus ask whether it is possible and useful to define an ordering in imaginary time. In the general case with operators A, B defined on the whole complex plane we could define an ordering in imaginary time G F z, A, B = T [Az 2Bz ] = { G + z, A, B if s < 0 G z, A, B if s > 0 39 to obtain an imaginary Feynman propagator, that is coincident with G on the strip,. Combining this with time ordering along the real axis becomes fairly abstract. In many applications it is favorable to define a path-ordering along a chosen contour in time instead of a global time ordering. 8

9 In conclusion we see, that in terms of our general argument the Feynman propagator plays no special role. It can be obtained by a special continuation prescription, but is in no way special in terms of the continuation to the complex plane. 3.2 Wick s theorem Using Wick s theorem we are able to compute interactions in perturbative quantum field theory by expressing them in terms of two-point functions. For the Feynman propagator to actually be useful we need to be able to use Wick s theorem in a thermal setting as well. In the following we are going to motivate how Wick s theorem can be carried over to the thermal case at finite temperature. The discussion mainly follows ref. []. Again we consider the complex scalar field. We look at the temperature expectation value of t operators S = α a, α b,2... α t,t = Tr e H α a, α b,2... α t,t 40 where α,µ = a µ, α 2,µ = a µ, α 3,µ = b µ, α 4,µ = b µ. First we note that [α a,, H] = [ ] ω µ α a,, a µa µ + b µb µ = ω µ λ a δ µ = λ a ω 4 µ µ with λ a = if a =, 3 and λ a = if a = 2, 4 and accordingly [ α a,, e H] = e λ aω. 42 By commutating α a, through all the other α, using the cyclicity of the trace and finally commuting α a, through the Boltzmann factor we obtain e λ aω S = Tr e H [αa, ] α c,3... α t,t, α b,2 + Tr e H [αa, ] α b,2... α t,t, α c, Tr e H [αa, ] α b,2... α s,s, α t,t. Using 27 one finds that [ ] e ω a µ µ, a δ µ = e ω = a µ a µ ] [a e ω µ µ, a = δ µe ω µ e ω = a µ a µ 44 and accordingly for b, b. Now dividing 43 by e λ aω the commutators on the right hand side become temperature expectation values. If we now apply this procedure multiple times we can decompose higher order functions into two point functions. This essentially illustrates how one can obtain the Wick theorem in a finite temperature case. Thus the set of tools familiar from ordinary quantum field theory carries over to the finite temperature case. 4 horizons and temperature In the following we derive the Unruh, Hawking and de Sitter temperature using the periodicity in imaginary time studied above. The presentation follows ref [2, 5, 3]. 4. Unruh temperature We consider a constantly accelerated observer in a Minkowski vacuum. 9

10 Figure 3: The Rindler wedge with lines of constant τ and ξ. Note that the variable z in the graph is referred to as x in the text. The region on the right is commonly referred to as the Rindler wedge. An accelerating observer experiences horizons at z = ±t. Taken from [3]. The metric in Minkowski space takes the form ds 2 = dt 2 dx 2 dx 2 45 where x are all directions perpendicular to the motion of the accelerated observer. If one now introduces new coordinates ξ, τ instead of x, t by the implicit definition setting the acceleration to one t = ξ sinh τ x = ξ cosh τ 46 one obtains a metric of the form ds 2 = ξ 2 dτ 2 dξ 2 dx The way these new coordinates parameterize the Rindler wedge is depicted in figure 3. If we now continue to imaginary time t = it E it does not matter whether we choose to rotate time in Rindler or Minkowski coordinates. By the coordinate mapping 46 continuing one of the times automatically implies, that we have to continue the other to imaginary times for consistency. After continuing to Euclidean time the metric takes the form ds 2 = dt E 2 + dx 2 + dx The coordinate transformations now are given by t E = ξ sin τ e x = ξ cos τ e 49 where τ = iτ e. Now due to 49 the time coordinate τ e has to be 2π periodic. This periodicity also becomes obvious when looking at the metric in Rindler space dropping an overall minus sign ds 2 = ξ 2 dτ E 2 + dξ 2 + dx 2, 50 0

11 which is the metric on a circle if τ E is 2π periodic, and otherwise would be singular at the origin. Identifying the period with = 2π we thus derive that the accelerated observer in Minkowski vacuum observes a temperature of T = = 2π. This derivation did not require a relation to general relativity. The existence of a horizon and the choice of hyperbolic coordinates for one observer already induces, that this observer will measure a temperature. 4.2 Hawking temperature In the case of Hawking radiation we consider the four dimensional Schwarzschild metric ds 2 = 2M dt 2 r 2M dr 2 r 2 dω 2 5 r where t, r are the time and radial coordinate and dω 2 is the usual line element on the sphere. Here M already contains Newton s constant such that M has units of inverse energy. The metric exhibits the famous coordinate divergence at the Schwarzschild radius r = 2M. Now we continue the time coordinate analytically to τ = it. This changes the metric to ds 2 e = 2M r = 2M r dτ 2 + 2M r 2M dr 2 + r 2 dω 2 r 2 52 dτ 2 + dr 2 + r 2 dω 2 again dropping an overall minus sign. To avoid the divergence of the line element at the Schwarzschild radius in imaginary time the bracket in the second line of 52 has to vanish when r 2M. Setting ε = r 2M we thus obtain the condition 0 = 2M 2 dτ 2 + dε 2 2M + ε τ 53 2 ε 2 d + dε 2 4M as ε 0. Upon rearranging and taking the square root one obtains Choosing the negative solution gives ±i dε ε = dτ. 54 4M ε e i 4M τ. 55 Thus the term in parenthesis in 53 vanishes if 4M is 2π periodic. We get = 8πM or T = 8πM, which is the result obtained by Hawking. Here we obtained the temperature by demanding that the metric at the horizon be regular. While this might not seem as straightforward as just introducing a coordinate change it is the natural thing to do. As there is no global time-like Killing vector field, there also is no natural definition of a vacuum. Instead one requires regularity of the metric at the horizon. Note that this solution is not stable. As M T = the temperature 8πM 2 rises if the mass of the black hole decreases. Thus even more power is emitted and the mass loss increases. Thus in the Schwarzschild case this solution is not stable. However it is a good approximation to the behavior of actual black holes, that have been formed by e.g. the collapse of a star. τ

12 4.3 De Sitter temperature Finally for de Sitter space the periodicity is relatively obvious. De Sitter space is a four-dimensional sub-manifold in a five dimensional Minkowski space defined by the condition t 2 i x 2 i = α2 56 where i =,..., 4 are the spatial coordinates, and α is an arbitrary constant. In cosmology de Sitter space is relatively important because it is the vacuum solution to Einstein s field equations with a cosmological constant. If we analytically continue the time variable in 56 by setting t = iτ we obtain τ 2 + i x 2 i = α2 57 and we are thus describing a sphere in five-dimensional Minkowski space. Accordingly τ just labels the circumference of this sphere and thus needs to be 2πα-periodic. We get a temperature of T = 2πα for de Sitter space. We have seen that the notion of temperature is observer dependent. In the Unruh case only the accelerated observer sees a finite temperature. Similarly in the case of Hawking radiation only the resting observer at infinity observes the Hawking temperature, a freely falling observer sees a locally Minkowskian spacetime and thus also observes Minkowski vacuum at zero temperature. With the KMS condition we can identify a temperature for every observer that observes a hyperbolic horizon. In conclusion, we find that the KMS condition provides us with an easy way to identify or define temperature in a quantum field theory. In the example of the complex scalar field we saw how the periodicity in complex time arose essentially due to the time evolution and the Boltzmann factor just differing by an imaginary unit. If for a certain observer a periodicity in complexified time is present, this can be identified with a temperature measured by this observer. In spacetimes with horizons this periodicity can be used as a tool to derive the temperatures associated with these horizons - for example Unruh, Hawking and de Sitter temperature. references [] T.S. Evans and D.A. Steer. Wick s theorem at finite temperature. Nuclear Physics B, 4742:48 496, 996. [2] Christopher J. Fewster. Lectures on quantum field theory in curved spacetime, [3] S. A. Fulling and G. E. A. Matsas. Unruh effect. Scholarpedia, 90:3789, 204. revision [4] Stephen A. Fulling and Simon N. M. Ruijsenaars. Temperature, periodicity and horizons. Physics reports, 523:35 76, 987. [5] Chethan Krishnan. Quantum field theory, black holes and holography. arxiv preprint arxiv:0.5875, 200. [6] Ryogo Kubo. Statistical-mechanical theory of irreversible processes. i. general theory and simple applications to magnetic and conduction problems. Journal of the Physical Society of Japan, 26: , 957. [7] Paul C. Martin and Julian Schwinger. Theory of many-particle systems. i. Phys. Rev., 5: , Sep