Fulling [ 11, and independently by Davies [3], who found that the Minkowski vacuum appears as a thermal state in the accelerated frame with temperature. (1.1).
ANNALS
OF PHYSICS 160,102-113
Thermal
(1985)
Radiation
from Accelerated
NATHAN
P.
Electrons
MYHRVOLD*
Department of Physics and Program Princeton University, Princeton,
in Applied Mathematics, New Jersey 08544
Received April 1, 1983
It is demonstrated that an electron in a constant external electric field will be in a thermal state and emit thermal radiation as a consequence of a non-perturbative effect in QED. The associated temperature is compatible with that found by Davies and Unruh for accelerated observers. Despite the thermal nature of the process, no evolution of pure states to mixed states is indicated. cs?1985 Academic Press, Inc.
1.
INTRODUCTION
One of the most interesting results from the study of quantum fields in curved space-time is the discovery that ordinary flat space quantum field theory displays strikingly new phenomena to accelerated observers. This was first demonstrated by Fulling [ 1, 21, who showed that the vacuum or no-particle state in the rest frame coordinates of a constantly accelerating observer is not unitarily equivalent to the ordinary Minkowski space vacuum. The Bogoliubov transformation coefficients between the accelerated and Minkowski frame “vacuum states” were calculated by Fulling [ 11, and independently by Davies [3], who found that the Minkowski vacuum appears as a thermal state in the accelerated frame with temperature
(1.1) where a is the magnitude of the proper acceleration of the frame and the units are such that A = c = k = 1. Unless otherwise specified, these units will be used throughout the remainder of this paper. The next major step was taken by Unruh [4], who considered the gedanken experiment of “flying” a model particle detector along a constantly accelerating or hyperbolic world line through the Minkowski vacuum. Unruh’s result was that the detector became excited in precisely the same manner as it would if it was immersed in a heat bath of radiation at the temperature given by (1.1). This explicitly demonstrates that the thermal distribution of particles found by Davies is a real physical effect rather than a mathematical artifact of the theory. * Supported by the Fannie and John Hertz Foundation, and in part by National Science Foundation PHYBO-19754. Current Address D.A.M.T.P., Silver Street, Cambridge, CB3 9EW, England.
Grant
102 0003.4916185 $7.50 Copyright 0 1985 by Academic Press, Inc. All rights of reproduction in any form reserved.
ACCELERATED
ELECTRONS
103
Considerable progress has been made toward the understanding of this phenomenon, most notably by Sciama [5] and his collaborators [6], who have presented an elegant interpretation in terms of the zero point fluctuations of the Minkowski vacuum state. Nevertheless, many important and fundamental questions remain unanswered. In particular, one can ask whether the Fulling-Davies-Unruh effect is consistent with ordinary flat space calculational methods, and if so, how can it be described in conventional terms? A related question concerns the observability of this effect; i.e., can Unruh’s model detector proceed to re-emit its excitations as thermal radiation which can be seen by nearby inertial observers? The prupose of this paper is to answer both questions by calculating the emission of radiation from an electron in a constant electric field. The classical motion of such an electron is simply the hyperbolic motion considered above, so an analysis based on canonical quantization in the electron’s rest frame leads one to believe that the electron will “see” a thermal bath of photons at temperature (1.1). One then expects the electron to “emit” radiation at this temperature by scattering the apparent heat bath to external inertial observers and to execute Brownian motion in its rest frame from the recoil. This heuristic picture is entirely confirmed by straightforward QED calculation of the emission of photons when the effects of the external field are treated nonperturbatively using Schwinger’s exact Green’s function [8]. An examination of the effect reveals that it is intrinsically non-perturbative in the external field. This paper is in four sections. Section II derives the emission spectrum of the electron, making extensive use of several prior results. The thermal nature of the radiation and the motion of the electron is exhibited in Section III. Finally, Section IV concludes the paper with a discussion of the result and its relevance to problems in curved space quantum field theory.
II. EMISSION OF RADIATION Before discussing the quantum mechanical aspects of an electron in a constant electric field, a brief mention of several facts must be made. The classical motion of a charged particle in a constant electromagnetic field has been treated at length in the literature [9, lo]. We shall choose coordinates x0 ( = t), x1, x2, x3 and let the field be a constant electric field oriented in the x3 direction. If the electron has no components of momentum transverse to the field, i.e., P,= (Pf + Pi)"* = 0, then the electron’s world line as a function of proper time r is xb(t) = (l/a)
[sinh(ar), 0, 0, cosh(at)] + xb(ro)
(2-l)
which is known as hyperbolic motion. The parameter a in Eq. (2.1) is the magnitude of proper acceleration and is a = etF/m, when P,= 0, where e and m are the
104
NATHAN
P. MYHRVOLD
magnitudes on the electron’s charge and mass, and & is the electric field strength. A sample world line of this form is shown for xb(rO) = 0 in Fig. 1. Two important features of the world line are readily apparent. First, the total energy is not a constant of the motion. An analogous conserved quantity is the “transverse energy” E, = (P: + m * ) ‘I* . Second, we expect that any physical process we wish to calculate will be independent of P, since changing P, simply acts as a time translation in a time independent field. Since the total time is infinite, we expect only to be able to calculate the emission rate per unit proper time. We may now turn to the quantum mechanical problem of obtaining solutions to the Dirac equation in a constant E field. The objective is to find electron wave functions which follow the classical motion above in the appropriate limit; however, care must be taken because the possibility of pair production in the E field gives rise to the famous Klein paradox. Narozhnyi and Nikishov [ 11-151 have solved this problem in two different ways. One is to work with a time dependent gauge, taking the vector potential A” to be Ab = [0, 0, 0, - at] and then selecting the appropriate solutions by asymptotic matching to the classical solution. This can be checked by using the vedtor potential Ab = [O,O, 0, -(2a/ar) tanh(at/2)] which dies off at co allowing one to define asymptotic states in terms of plane waves. In the adiabatic limit as the “dummy” parameter a + 0, both methods yield the same orthonormal set of solutions to the Dirac equation. Another method is to use a space dependent gauge and treat the calculation as a barrier penetration problem [ 151. If we denote the sign of the frequency of the solutions at t = +co with superscripts as Vi, or -f, and those at r = -co by subscripts as +f,,, f,, then, following Narozhnyi and Nikishov [ 13, 141 we have rf,,,
= exp{ - n/(87r) + ip . x}(2VegI)-“* x bA,*Wl
- i)) + (v)Gw’*
(1 - 9 qi,,,,-,w(l
- Q)l
Tfp,n = exp{ - 1/(87r) + ip . x}(4VeZ)-“* X [z@-Cj~,2,-l(r>A2 the wavefunctions fp,n are peaked about the classical world line discussed above. When r2 N A, quantum mechanical effects become important and pair production can occur. This will be discussed in more detail later in the paper. The causal or Feynman Green’s function can be expressed in terms of these modes as
G,(x,x’)= W’- 1)c +f,N, P
t?(t - t’) x -fp(x’)
-f,(x) N,
(2.4)
P
where 8 is the step function and the normalization factor N, is
N,=
(-1/2)(1/rr)“2
T(U/2) exp{&/4}.
The process we wish to examine is the emission of photons by an electron. To the lowest order of perturbation theory in the electron-photon coupling, this is represented by the Feynman graph in Fig. 2, where the double lines are a reminder that we are using the Schwinger Green’s function (2.4) rather than the usual free Green’s function. The amplitude for the graph of Fig. 2 has been evaluated by Nikishov [ 161. After some manipulation one obtains the differential probability for the emission of a photon with frequency k, = w and momentum k which, when
106
NATHAN P.MYHRVOLD
FIG. 2. Feynman graph for photon emission in non-perturbative external field.
summed over the polarizations has the form dw = d3k(w)-’
of the photon and the final spin states of the electron,
exp{ - $A -A’)}
X [ 167r28*(1 - exp{ - nlz’} sinh(n1/2)]-’
+ (A x n’)(u’)-’
{ ]l - m2/(e81Z’)] e8 1Yy12
k: 1Y’ I2
- [(P,k,)2/(ebM’)
+ (1 - L’/A)(l - ky(egA’))
es] Re{ !@P}}
where, k;=k;+k; 1’ = E;‘/e = (ea)-’
[(P1 - kJ2 + (P2 - k,)’ + m2].
G-w
The symbol Y denotes the confluent hypergeometric function of the second kind; written in full these are Y = Y(u, c; z) Y’ = dY/dz and in (2.5) above a = U/2,
c = 1 + (i/2)0, - A’),
z = - (i/2&)kt.
Although Eq. (2.5) is rather formidable, several of its features are readily apparent. As expected from the discussion above, Eq. (2.5) is independent of P,. Since k, , the x3 component of the momentum in the inertial or “laboratory” frame, depends upon the value of P,, it also acts as a time translation and does not appear in (2.5). It should be emphasized that this property is an artifact of the field being time independent, and is also true for the case of classical radiation [ 171.
III.
THERMAL
PROPERTIES
Much of the complexity of Eq. (2.5) has its origin in the fact that it describes photons in a laboratory or rest frame which is comoving with the electron at t = P,/eZ’. It is therefore much easier to examine its properties by studying the behavior of the electron. In the emission process the electron goes from having an initial transverse energy
ACCELERATED
E, to a final value El. emission is
In terms of these values, the differential
dw = ~f~k(27r)-~ n(e/oa)
exp{-(n/e8)[(1/2)
[l -exp(-E*Jeb]]-‘[l
x
107
ELECTRONS
E:-
- 2m2) ( !?I2 + (k,/EJ2(E:+
- E;‘[k;(E;
+ El2 - 2m2) + E:-
for
EL’])
-exp{-E’:/e8~]-’
x {(E:+E12
probability
Elp2 EL2) ( Y’(*
(3.1)
El41 Re{ YY}}.
Inspection of Eq. (3.1) shows that the electron may either gain or lose transverse energy upon emission since dw is non-zero for either sign of (Et - E’,Z). Since we expect a thermal distribution for the transverse energy, we can look at the ratio of the probability for the electron to gain an amount of transverse energy 6 = EL - E, by emitting a photon with a given k: to the probability for losing an amount with the same photon momentum. If we let C&V* represent dw after the transformation interchanging, E I -+ E 1, E ‘1.-+ E I, then probability of gaining 6 dw W4 Pr( - 8) = probability of losing 6 = dw*
(3.2)
In order to relate the two we can use several identities for confluent hypergeometric functions (see Appendix A). Let Y= Y(a, c; z), Y* = Y(u*, c*; z) then, !PF=
exp{ - (r/2eg)(E:
Y/‘Y’=exp{
-(?r/2ea)(Et-E>*)}[Y’*p’*
+ kL4(El
- EL2) Y*p*
Re{ Y’F] = exp{ - (z/2eZ’)(E:x [Re{ Y’*Y*} Upon substitution
(3.3a)
- E12)} Y*F*
- EL2) Re{ Y’*Y*}]
- 2k;‘(E:
E12))
- k;‘(Et-
(3.3b)
(3.3c)
E12) Y*F*].
into dw in Eq. (3.2), the above relations tell us that dw/dw* = exp{-(n/eg)(E:-
El’)}.
In terms of the energy difference 6 this is just a Boltzmann Pr(G)/Pr( - S) = exp{ - 6/T}
(3.4) factor (3.5)
where the effective temperature T is T= e8[z(E,
+ E’,)]-’
= e8(27cEl,,)-’
(3.6)
and Elav = (1/2)(E, + E’,) is the average transverse energy. Equation (3.6) is highly reminiscent of temperature (1.1) found by Davies [3] and Unruh [4], and reduces to it when the transverse momentum of the electron is small
108
NATHAN
P. MYHRVOLD
compared to its mass. This is not surprising since temperature (1.1) was derived for a system constrained to follow a world line with P, = 0. Equation (3.5) has important consequences for the motion of the electron since it prescribes the power spectrum of fluctuations in the electron’s transverse momentum. In the rest frame of an observer which follows the classical world line of Fig. 1 (with P, = 0) the electron will execute Brownian motion. This behavior is reminiscent of Einstein’s famous gedanken experiment of a mirror moving in a box filled with thermal radiation [ 181. The primary significance of the Boltzmann factor in (3.5) is that it demonstrates that the electron acts precisely as if it was in equilibrium with a heat bath at temperature (3.6). Since the changes in the electron’s transverse energy are exactly correlated with the emission of photons, we expect that the spectral distribution of the photons is also thermal at temperature (3.6). This is not manifestly obvious from an inspection of (2.5) or (3.1) because of several factors which confuse the issue. As discussed previously, (2.5) and (3.1) are based in the “laboratory” frame, so the relevant temperature is related to (3.6) by a Lorentz transformation. Since dw is also a time average over the entire range of the electron, the effective temperature seen is smeared out by averaging over the time dependent Lorentz factor. (See, however, the discussion in Section IV.)) The main difficulty with (2.5) and (3.1), however, is that they include radiation effects which are unrelated to the changes in E I, such as remnants of the classical radiation spectrum and the effects of the electron’s spin flipping upon emission. Since these effects are described by quite complicated expressions themselves [ 12, 171, one cannot expect to just subtract them from (2.5) or (3.1) without getting another intractable expression. One of the great advantages of Eq. (3.2) is that is allows us to weed out these complicated factors. The spectral distribution of the component of the radiation which is associated with changed in E is closely related to the issue of whether this process requires pure states to go to mixed states. The thermal nature of the electron’s energy would seem to suggest this while, on the other hand, the laws of quantum mechanics require just the opposite. The apparent paradox is resolved if one recalls that the thermal fluctuations of the electron are exactly correlated with the fluctuating emission of photons. The combined system of the electron and the photons it emits (in this mode of emission) is always a pure state globally. Local observers who choose to ignore the correlations by looking only at the electron or only at the photons will however find that they are indistinguishable from thermal states. A detailed interpretation of how this arises from interactions with vacuum fluctuations can be found in Refs. [5,6]. IV.
DISCUSSION
It is interesting to compare the results of Sections II and III with the usual QED perturbation theory done in the absence of an external field. The lowest order
ACCELERATED
ELECTRONS
109
Feynman graphs for the emission of a photon by an electron are then given by Fig. 3. The corresponding amplitude for emission is well known, and is equivalent to the classical expression for radiation from an accelerating charge. Despite the fact that the graphs in both Figs. 2 and 3 appear to describe “tree level” processes, the external field propagator (2.4) in Fig. 2 contains non-perturbative quantum information about the theory not found in graphs like those of Fig. 3, with any number of external photon lines. It is also clear that the processes cannot be represented by radiative corrections to the tree level process, such as the graphs in Fig. 4, which are some of the l-loop corrections to the graphs of Fig. 3. This becomes apparent when we restore the physical constants to Eq. (3.5) to obtain Pr(J)/Pr(--6)
= exp{-(2xE~,,)/(AceZ)}.
(4.1)
The exponential dependence on h found in Eq. (4.1) is symptomatic of a nonperturbative process since a loop expansion of (4.1) in powers of h yields a series with all coefficients equal to zero. Christenson and Duff [21] have demonstrated that one can interpret the Fulling-Davies-Unruh phenomena as an “instanton” effect in the geometry and topology of flat space, so the non-perturbative nature seen here comes as no real surprise. In fact, the most famous example of a non-perturbative effect in QED, pair creation in an external field, was first calculated [8] using a propagator equivalent to Eq. (2.4) used in Fig. 2 and the calculation above. The pair creation probability has been also found by methods directly related to those used in this paper [ 12). The total probability for producing an electron-positron pair with transverse momentum P, is Wp, = exp{-@/(ea)}.
(4.2)
Following Eq. (3.5) this can be interpreted as a Boltzmann factor albeit at a temperature twice that found for (3.5). Frolov and Gitman [22] have shown that if one examines the density matrix coresponding to either the electrons or the positrons alone then it will be a thermal density matrix. This is just another example of the fact that ignoring correlations can make vacuum fluctuations look like thermal fluctuations. The results derived in this paper have been obtained for the unbounded motion of an electron in a constant E-field of infinite extent. On physical grounds, however, one would expect that an electron which was in a constant field for a time long compared
FIG. 3. Graph for photon emission with perturbative treatment of external field.
110
NATHAN
FIG.
P. MYHRVOLD
4. Examples of l-loop corrections to the process of Fig. 3.
to some critical time scale would have an emission spectrum which closely approximated (2.5) or (3.1). This is indeed the case, as can be shown by a steepest descents approximation to the integrals from which Eq. (2.5) was obtained. The characteristic proper time interval is approximately given by temperature (3.6) At N T 2: l/u. As a numerical example, it happens that if the acceleration of the electron is equal to lg (the surface gravity on the earth), then t is almost exactly one year. Thus if an electron stays in a field producing a proper acceleration of lg for a time large compared to a year, we would expect to find it in the thermal state predicted by Eq. (3.5) although in this example the resulting temperature would be incredibly small (on the order of lo-i9 degrees Kelvin). The photon emission process has been considered here only at the lowest order in perturbation theory with regard to the electron-photon coupling. Higher order tree level processes and radiative loop corrections may well be important for certain specific applications, but are unnecessary to demonstrate the connection between external field QED and the Fulling-Davies-Unruh effect which is the primary focus of this paper. An interesting example of a higher order process is provided by the graph of Fig. 5. The amplitude for this process in a constant electric field has been calculated by Baier and his co-workers [23], but a qualitative understanding can be obtained by inspection. We know from the results of Sections II and III that the process of Fig. 2 will produce a flux of photons. On the other hand, it is well known [ 12, 131 that any photon in an external field is unstable to pair production, so we may describe Fig. 5 as pair creation by an accelerating electron. Since each particle in the pair, as well as the original electron, will individually go off and repeat the process, this leads to a runaway production of particles at an exponentially increasing rate. This behavior is very important as a model of what might happen in another external field problem of interest-particle production in curved space times, where gravity plays the role of the external field. This connection is not merely formal since the Fulling-Davies-Unruh thermal effects, which have been demonstrated here for an electron falling in a constant electric field, are also present for particles or model “particle detectors” falling in the homogeneous gravitational field of the de Sitter universe [5,6,24]. Recent calculations [25] indicate that a runaway particle production process does exist for lq4 theory in de Sitter space, and general arguments
111
ACCELERATEDELECTRONS
FIG. 5.
Graph
for pair production
by an electron
in an external
field.
suggest that QED in de Sitter space will possessdirect analogs of the graphs in Figs. 2 and 5. These particular production mechanismshave enormous implications for the dynamics of the very early universe and possibly for our understanding of the cosmological constant [26]. The external field QED calculations presented in this paper provide a “down to earth” model of the physics responsible for the cosmological particle creation which does not suffer from many of the conceptual uncertainties which plague the curved space quantum field theory problem. The Fulling-Davies-Unruh effect is also closely related [5,6] to Hawking radiation from black holes [27], so the results of Section III also lend credence to the consistency of Hawking radiation with conventional flat space QED. Perhaps the most striking aspect of the relationship between the curved space quantum field theory effects and the accelerating electron radiation considered here is that the latter case lends itself to experimental verification in the forseeable future. Beyond the obvious experimental interest in observing the thermal radiation as a nonperturbative QED effect, this process can also be viewed as a direct laboratory example of the same physics which is behind black hole evaporation and particle production in the very early universe. The question of whether or not this Fulling-Davies-Unruh thermal radiation can be observed experimentally has been the subject of speculation by several authors [28, 291. These calculations, however, were of necessity based on simple dimensional arguments and order of magnitude estimatesusing the Davies and Unruh results. The results of Section III confirm that the thermal radiation does in fact exist within the context of orthodox flat space theory, and at the sametime they provide an accurate assessmentof its properties for use in the design of practical experiments. Recent investigation [30] suggests that the experimental situation is hopeful. Detailed calculations based on external field QED are a subject of continuing investigation.
APPENDIX
A
In order to relate the confluent hypergeometric functions Y= Y(u, c; z) and Y* = IY(a*, c*; z) with,
a = iE:/2e8,
a*=iEi2/2e8,
c=l+a--a*,
z = -ik2J2e, c*=l+a*-a
(A.1)
112
NATHAN P.MYHRVOLD
we may use Kummer’s
transformation
(see [3 1, p. 1211)
Y(t2, c; 2) = Z’-c Y(a - c + I,2 - c; z)
(A.2)
for any complex a, c, z. Using (A.l) and (A.2) this directly yields Y = Y* exp{ - (Et - E;‘)(2e8)-‘[rr/2
- i log(-z)]}.
(A.3 )
The function Y’ = dY/dz must first be written in terms of Y functions using the identity (see [32, p. 2581) (d/dz) Y(t.2,c; z) = a qa + 1) c + 1; z). We may then apply Kummer’s
transformation
64.4)
(A.2) and the identity
(tf/dz) Y(tz, c; z) = z(c - a - 1))‘(d/dz)
Y((a, c; z)
+ z-‘(c - 1) Y(a, c;z)
(A.5 )
to obtain Y’ = exp{-(El
-E;*)(2eg)-‘[Z/2
x [Y’* - k;*(E;Equations (3.3a)-(3.3c)
- i log(-z)]} (A4
El*) Y*].
can be directly obtained from (A.3) and (A.6).
ACKNOWLEDGMENTS I would like to thank Malcolm Perry, Gregory Adkins, Charles Whitmer and Steven Bottone for many stimulating conversations. This work was supported by a fellowship from the Fannie and John Hertz Foundation, and in part by National Foundation Grant PHY80-19754.
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