Beyond flat-space quantum field theory

37 Pages·2000·0.462 MB·English

by  Seahra S.

Beyond flat-space quantum field theory
Sanjeev S. Seahra
Department of Physics
University of Waterloo
May 11, 2000
Abstract
We examine the quantum field theory of scalar field in non-Minkowski space-
times. We first develop a model of a uniformly accelerating particle detector and
demonstrate that it will detect a thermal spectrum of particles when the field is in
an “empty” state (according to inertial observers). We then develop a formalism for
relating field theories in different coordinate systems (Bogolubov transformations),
and apply it to compare comoving observers in Minkowski and Rindler spacetimes.
Rindler observers are found to see a hot bath of particles in the Minkowski vac-
uum, which confirms the particle detector result. The temperature is found to
be proportional to the proper acceleration of comoving Rindler observers. This
is generalized to 2D black hole spacetimes, where the Minkowski frame is related
to Kruskal coordinates and the Rindler frame is related to conventional (t,r) co-
ordinates. We determine that when the field is in the Kruskal (Hartle-Hawking)
vacuum, conventional observers will conclude that the black hole acts as a black-
body of temperature κ/2πk (k is Boltzmann’s constant). We examine this result
B B
in the context of static particle detectors and thermal Green’s functions derived
from the 4D Euclidean continuation of the Schwarzschild manifold. Finally, we give
asemi-qualitative2Daccountoftheemissionofscalarparticlesfromaballofmatter
collapsing into a black hole (the Hawking effect).
1 Introduction
Thespecialtheoryofrelativitypostulatesthatallinertialreferenceframesareequiv-
alent. That is, the laws of physics are symmetric under the Lorentz group, which
consists of all the proper Lorentz rotations. In quantum field theory, one usually
makes the additional demand that physical systems be invariant under four di-
mensional translations, which has the net result of making the Poincar´e group a
symmetry of fundamental interactions. In other words, quantum fields look the
same to all inertial observers. However, these symmetries are far too restrictive if
quantum field theory is to be understood in the sense of general relativity. The
principle of general covariance asserts that the laws of physics be invariant under
arbitrary coordinate transformations. All observers are to be treated on an equal
footing, regardless of how they are moving.
This paper attempts to examine what changes in the standard formulation of
quantumfieldtheorywhenoneallowsforarbitrary,non-inertialobservers. Webegin
by constructing a simple model of a particle detector moving along an arbitrary
world line x(τ) and discover that the particle content of the “vacuum” is entirely
dependent on the state of motion of the detector, which is the so-called “Unruh
effect”. This motivates us to study the notion of the vacuum in more detail, which
leads to the conclusion that the ground state as defined in Minkowski coordinates
|0 (cid:105) need not be the same as in arbitrary coordinates. To quantify the difference
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between field theories in difference coordinate systems, we introduce Bogolubov
transformationsbetween“plane-wave”expansionsofquantumfields. Were-examine
the case of accelerating observers by studying the Rindler spacetime where spatially
comoving observers are in fact uniformly accelerating. The black hole case is then
considered in analogy to the Rindler spacetime, and the temperature of external
black holes is derived. We re-derive this result using a static particle detector model
andthermalGreen’sfunctionsderivedfromtheEuclideananalogueofSchwarzschild
space. We end off by giving a semi-qualitative account of the Hawking radiation
emitted from a collapsing ball of matter in 2D and black hole evapouration. We will
be using the standard metric signature of quantum field theory (+−−−), and will
often work with two dimensional models.
2 Accelerating particle detectors
In this example, we will consider a scalar field propagating in a 1+3 spacetime1.
The generalization to higher spins, while straightforward, would just clutter the
notion and cloud the ideas. We begin by constructing a model for a detector that
can be used to determine if there are any particles in a given quantum state of
the field φ(x). Our particle detector will be a quantum system with energy levels
{E }∞ and a non-interacting Hamiltonian H . The standard field Hamiltonian
n n=0 d
1We follow section 3.3 in Birrell & Davies [1] and Unruh’s paper [2].
1
will be denoted by H . The detector will move along a worldline x(τ), where τ is
f
the proper time, and it will be coupled to the scalar field via a (small) monopole
momentoperatorm (thesubscriptindicatesthatwearedealingwithaSchr¨odinger
S
operator). In the Schr¨odinger representation, the interacting Hamiltonian is
H(τ) = H +H +m φ(x(τ)). (1)
f d S
H(τ) reduces to H ≡ H +H if the coupling m goes to zero. The eigenstates of
0 f d s
H are given by
0
|k,n(cid:105) ≡ |k(cid:105)⊗|n(cid:105), (2)
H |k(cid:105) = ω |k(cid:105), (3)
f k
H |n(cid:105) = E |n(cid:105), (4)
d n
(cid:112)
where ω = |k|2+m2. The ground state of the H operator may be written
k 0
as |0 ,0(cid:105), where |0 (cid:105) is the standard Minkowski vacuum field configuration. We
M M
suppose that in the distant past τ → −∞, the system is in the ground state. We
wish to calculate the probability amplitude that the system will be found in another
eigenstate|k,n(cid:105)ofH atsomelatertimeτ. Ifanobservertravelingwiththedetector
0
initiallypreparesthedeviceinthegroundstateinthedistantpastandmakesanon-
zero measurement of the energy in the future, she will conclude that the detector
absorbed energy from the field. That is, she will have detected a particle excitation
of the field. Because the Hamiltonian is an explicit function of time, we do not
expect energy to be conserved in this system.
Wewillcalculatetherequiredprobabilityamplitudetofirstorderinthemonopole
moment m . It is easiest to first work in the Schr¨odinger picture and then partially
s
convert the result into the Heisenberg form . The state vector at some arbitrary
time |ψ(cid:105) can be expanded in terms of eigenstates of H :
τ 0
(cid:88)
|ψ(cid:105) = c (τ)|k,n(cid:105) , (5)
τ k,n τ
k,n
where we have chosen a box normalization. The probability amplitude of measuring
the state of the system to be |k,n(cid:105) at some time τ is c (τ). H(τ) governs the time
k,n
evolution of |ψ(cid:105) while H governs the time evolution of |k,n(cid:105) via their respective
τ 0 τ
Sch¨odinger equations. Now, we can take an explicit time derivative of |ψ(cid:105) to get
τ
(cid:88) (cid:88)
−i (H +m φ )c |k,n(cid:105) = (c˙ −iH c )|k,n(cid:105) , (6)
0 S s k,n τ k,n 0 k,n τ
k,n k,n
where we have indicated that φ = φ (x(τ)) is to be understood as a Schr¨odinger
s s
operator. Also, c˙ = dc /dτ. Taking the inner product with (cid:104)p,r| and making
k,n k,n τ
use of orthonormality gives
(cid:88)
c˙ = −im c [ (cid:104)p,r|φ |k,n(cid:105) ]. (7)
p,r S k,n τ s τ
k,n
2
Now, we replace the Schr¨odinger vectors by their Heisenberg counterparts using
|k,n(cid:105) = e−i(ωk+En)τ|k,n(cid:105) = e−iEnτe−iHfτ|k,n(cid:105), (8)
τ
and
(cid:104)p,r| = (cid:104)p,r|e+i(ωp+Er)τ = (cid:104)p,r|e+iErτe+iHfτ. (9)
τ
However, to zeroth order in m
S
φ(x(τ)) = eiHfτφ (x(τ))e−iHfτ, (10)
s
since φ (x(τ)) must commute with H . Now, the zeroth order solution to (7) corre-
s d
sponding to the initial condition that |ψ(cid:105) is the ground state is
τ
(0)
c = δ δ . (11)
p,m p,0 m,0
Puttingthiszerothordersolutioninto(7)andintegratingwithrespecttotimeyields
our final result
(cid:90)
τ
c(1)(τ) = −i(cid:104)n|m |0(cid:105) dτ(cid:48)ei(En−E0)τ(cid:48)(cid:104)k|φ(x(τ(cid:48)))|0 (cid:105). (12)
k,n S M
−∞
for k (cid:54)= 0 and n (cid:54)= 0. The matrix element ξ ≡ −i(cid:104)n|m|0(cid:105) depends on the details of
n
the detector structure, and will hence remain unspecified.
Equation (12) represents the probability amplitude that the system will make
a transition from the ground state to the excited state at some arbitrary time τ.
What is the probability that there will be one particle of 4-momentum kα in the
final state? It’s straightforward to calculate
(cid:104)1 |φ(x(τ))|0 (cid:105) = e+ik·x(τ), (13)
k M
using the standard expansion for φ(x), |1 (cid:105) = a†|0 (cid:105) and [a ,a† ] = δ in the
k k M k k(cid:48) k,k(cid:48)
box normalization. Let’s first consider the case when the detector is moving along
an inertial (non-accelerating) trajectory,
τ (1,v)
xα(τ) = √ , (14)
1−v2
where v is a constant vector such that v2 = v·v < 1. Hence.
(cid:90) (cid:189) (cid:183) (cid:184) (cid:190)
τ ω −v·k
c(1) (τ) = ξ dτ(cid:48)exp i (E −E )+ √k τ(cid:48) (15)
1k,n n n 0 1−v2
−∞
Taking the limit as τ → ∞ we get a delta-function,
(cid:183) (cid:184)
ω −v·k
c(1) (∞) = 2πξ δ (E −E )+ √k . (16)
1k,n n n 0 1−v2
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However,E > E andω > |k||v| > k·v. Hence,theargumentofthedeltafunction
n 0 k
is strictly positive and the transition is forbidden on energy grounds. Therefore,
inertial observers will measure the particle content of the field to be zero in the
distant future, just as it was in the distant past.
How about non-inertial observers? It’s easier in this case to calculate the prob-
ability P that the detector will be found in the nth eigenstate as τ → ∞:
n
(cid:88)
P = |c(1)(∞)|2
n k,n
k
(cid:90)∞ (cid:90)∞
= |ξ |2 dτ dτ(cid:48)e−i∆E(τ−τ(cid:48))(cid:104)0 |φ(x(τ))φ(x(τ(cid:48)))|0 (cid:105), (17)
n M M
−∞ −∞
(cid:80)
where ∆E ≡ E −E and we have made use of 1 = |k(cid:105)(cid:104)k|. Now,
n 0 k
(cid:88) (cid:104)0 |a e−ip·x(τ)a†e+ik·x(τ(cid:48))|0 (cid:105)
(cid:104)0M|φ(x(τ))φ(x(τ(cid:48)))|0M(cid:105) = M p √ k M
2V ω ω
k p
k,p
(cid:88) e−ip·x(τ)e+ik·x(τ(cid:48))δ
k,p
= √
2V ω ω
k p
k,p
(cid:90)
1 d3k
→ e−ik·[x(τ)−x(τ(cid:48))]
(2π)3 2ω
k
= i∆+[x(τ)−x(τ(cid:48))], (18)
where the limit in the third line is taken for V → ∞ and ∆+(x) is the standard plus
Green’s Function for the Klein-Gordon field, also known as the Wightman Green’s
function. Let’s specialize to the massless case where,
i 1
∆+(x) = D+(x) = . (19)
(2π)2(x−iη)2
This expression is to be understood in the limit of η → 0, where ηα = (η,0,0,0) is
a small future pointing vector.
Let’s evaluate the Green’s function for the inertial path (14). We have
√
(τ −τ(cid:48)−iη 1−v2)2−(τ −τ(cid:48))2v2
[x(τ)−x(τ(cid:48))−iη]2 =
1−v2
= (τ −τ(cid:48)−i(cid:178))2,
where (cid:178) = η/(1−v2)1/2. Then, (17) becomes
(cid:90)∞ (cid:90)∞
|ξ |2 e−i∆Eζ
n
P = − dτ dζ , (20)
n (2π)2 (ζ −i(cid:178))2
−∞ −∞
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where we have made the change of variable ζ = τ −τ(cid:48). This expression is formally
infinite because of the integration over τ, but may be regulated by defining the
(cid:82)
∞
transition probability per unit time as p = P / dτ. Since ∆E > 0, we perform
n n −∞
the ζ-integral by completing the contour in the lower-half plane. But the only pole
on the integrand is at ζ = i(cid:178) in the upper-half plane which means the integral is
zero. Hence, p = 0 for inertial observers, confirming our previous result that such
n
observers do not detect any particles.
Now, consider a detector following a hyperbolic trajectory
xµ(τ) = α[sinh(τ/α),0,0,cosh(τ/α)]. (21)
√
It’seasytoverifythatthemagnitudeofthedetector’sproperaccelerationis aµa =
µ
α−1, where aµ = d2xµ/dτ2. That is, the detector’s acceleration is measured to be a
constant in an instantaneously comoving frame. Now,
(cid:183) (cid:181) (cid:182) (cid:184)
(cid:179)τ(cid:180) τ(cid:48) 2
[x(τ)−x(τ(cid:48))−iη]2 = α2 sinh −sinh −iη
α α
(cid:183) (cid:181) (cid:182)(cid:184)
(cid:179)τ(cid:180) τ(cid:48) 2
−α2 cosh −cosh
α α
(cid:181) (cid:182)(cid:181) (cid:182)
τ −τ(cid:48) i(cid:178) 2
= 4α2sinh2 1−
2α α
(cid:181) (cid:182)
τ −τ(cid:48) i(cid:178)
= 4α2sinh2 − , (22)
2α α
where (cid:181) (cid:182)
τ +τ(cid:48)
(cid:178) = αηcosh2 . (23)
2α
Then, p becomes
n
(cid:90)∞ (cid:181) (cid:182)
|ξ |2 ζ i(cid:178)
p = − n e−i∆Eζcsch2 − dζ. (24)
n (4πα)2 2α α
−∞
Now,
(cid:181) (cid:182)
ζ −2i(cid:178) (cid:88)∞ 4α2
csch2 = (25)
2α (ζ −2i(cid:178)+2πikα)2
k=−∞
Putting (25) into (24) yields
(cid:90)∞
|ξ |2 (cid:88)∞ e−i∆Eζdζ
n
p = − . (26)
n 4π2 (ζ −2i(cid:178)+2πikα)2
k=−∞−∞
Now, as mentioned above, to do the integral we need to complete the contour in the
lower-half plane. However, each term in the integrand has a second-order pole at
5
ζ = i(2(cid:178)−2πkα). Hence, there will only be contributions for k = 1...∞. For those
values of k, the 2i(cid:178) term is irrelevant and we can use residue theory to get
(cid:90)∞
e−iβζdζ
= −2πβe−iβγ, β,γ > 0, (27)
(ζ +iγ)2
−∞
and
|ξ |2 E −E
n n 0
p = . (28)
n
2π e2πα(En−E0)−1
This is the probability per unit time that the detector will absorb an amount of
energy E −E from the “vacuum”-state |0 (cid:105). We can interpret the |ξ |2(E −E )
n 0 M n n 0
term as the sensitivity of the detector at this particular energy. But the probability
p should go like the sensitivity of the detector to particles of energy E −E times
n n 0
the relative number of such particles in the field, which in this case would be given
by
1
. (29)
e2πα(En−E0)−1
Comparethistotheexpressionfromstatisticalmechanicsfortheoccupationnumber
of the energy levels between E and E +dE when a gas of bosons is in equilibrium
with a heat bath of temperature T:
1
, (30)
e(E−µ)/kBT −1
where k is Boltzmann’s constant. The similarity of the two expression leads us to
B
conclude that an observer traveling along with the particle detector will conclude
that she is moving through a hot gas of bosons with a temperature of
T = 1/2παk . (31)
B
When she makesthe identificationE = E −E , she will conclude that the chemical
n 0
potentialµofthegasiszero. Thatis,thebosonsaremassless. Theconclusionisthat
uniformly accelerating observers will not see the quantum field in it’s ground state,
but will rather see a thermal excitation of the field with a temperature proportional
totheiracceleration. Itshouldbeclearthatenergyisnotconservedinthissituation
because the initial state has E = E while the final state has E > E . Where
0 0
did the energy come from? The standard answer is that the agent responsible for
accelerating the detector must do work on the field φ. Then, when the detector
interacts with the field, the energy is used to excite the detector out of the ground
state.
3 Bogolubov Transformations
In reading the previous section, the reader might wonder whether or not the fact
that the uniformly accelerating observer detects a bath of thermal bosons depends
6
on the details of the detector model adopted. It turns out that the conclusions
are independent of the detector, and are rather based in the difference between the
natural coordinate systems used by inertial and non-inertial observers2.
To make the last statement more concrete, let us consider a flat 1+1 spacetime
with the line element
ds2 = dt2−dx2. (32)
We have suppressed the two spurious spatial dimensions found in the last section.
Thecoordinatesystemrepresentedby(32)iswellsuitedtoinertialobserversbecause
wecanalwaysuse2DLorentztransformationssuchthatanyinertialobservermoves
on a x = constant trajectory. Consider the transformation
t = a−1eaξsinh(aη)
x = a−1eaξcosh(aη), (33)
which casts the line element in the form
ds2 = e2aξ(dη2−dξ2), (34)
which is the defining relation for the 2D Rindler spacetime. The coordinate system
represented by (34) is well suited to observers moving on ξ = constant trajectories,
given by
xµ(η) = (t(η),x(η)) = a−1eaξ(sinh(aη),cosh(aη)). (35)
But the proper time for ξ = constant observers is τ = eaξη. Hence, by compar-
ing (21) with (35), we conclude the spatially comoving observers in the Rindler
spacetime are uniformly accelerating with a proper acceleration α−1 = ae−aξ.
We can reasonably assume that observers will tend to construct quantum field
theories in coordinate systems where they are comoving; that is, in their own rest
frames. So, inertial observers will attempt to formulate a quantum description of
the field φ in the (t,x) coordinate system, while uniformly accelerating observers
will attempt to do the same in the (η,ξ) system. The question is, how are the two
representations related?
The answer is given in terms of Bogolubov transformation between the “plane-
wave” decompositions of φ in different coordinate systems. Although we will return
to the Minkowski and Rindler spacetimes in the next section, we now work in a
general curved manifold of dimension n. We will assume that the spacetime admits
the existence of a timelike Killing vector field, which will allow us to make a sen-
sible definition of positive frequency modes. We also demand that the manifold be
globally hyperbolic, which makes the initial value problem for the field φ tractable.
The relativistic generalization of the Klein-Gordon equation is
(cid:161) (cid:162)
∇α∇ +m2+ζR φ = 0, (36)
α
2We draw on section 3.2 of Birrell & Davies [1] and section 14.2 of Wald’s relativity text [3] for
the discussion in this section.
7
where m is the mass, R is the Ricci scalar and ζ is a constant that defines the
coupling of of the field to the curvature of the manifold. The ζ = 0 case is referred
to as minimally coupled, while the ζ = [(n−2)/(n−1)]/4 case is referred to con-
formally coupled because the massless wave equation is invariant under conformal
transformations (g → Ωg ,φ → Ω1−n/2φ). In 2D, the minimal and conformal
αβ αβ
coupling cases coincide, which allows for considerable simplification in the solution
of the wave equation for conformally flat spaces.
The wave equation will in general involve a number of mode solutions {u ,u∗}
i j
which are eigenfunctions of the Lie derivative operator
£ u = −iω u (37)
ξ i i i
£ u∗ = +iω u∗, (38)
ξ i i i
where ξα is a timelike Killing vector andω > 0. The label i is used to schematically
i
tell the difference between modes and may be continuous or discrete. The modes
{u } are said to be of positive frequency, while the modes {u∗} are of negative
i i
frequency. We define the scalar product between two functions as
(cid:90)
−→ ←−
(φ ,φ ) = −i φ (x)(∂ −∂ )φ∗(x)dΣα (39)
1 2 1 α α 2
Σ
Where Σ is a spacelike (Cauchy) surface. On can show that the value of the scalar
product (φ ,φ ) is independent of the surface Σ. Also, note that
1 2
(φ ,φ ) = −(φ∗,φ∗). (40)
2 1 1 2
The mode solutions {u ,u∗} are orthonormal in the sense
i j
(u ,u ) = δ , (u∗,u∗) = −δ , (u∗,u ) = 0. (41)
i j ij i j ij i j
Inn-DMinkowskispace,itiseasytoverifythatthemodesolutionstotheminimally
coupled wave equation
(cid:112)
u (x) = [2ω (2π)n−1]−1/2e−iωkt+k·x, ω = |k|2+m2, (42)
k k k
satisfy the relations (37) and (41) in the limit where k is a continuous label, with
ξ = ∂/∂t and Σ equal to a surface of constant time. To achieve quantization of the
field, we expand the field operator φ(x) in terms of the mode functions {u ,u∗}
i j
(cid:88)(cid:104) (cid:105)
φ(x) = a u (x)+a†u∗(x) , (43)
i i i i
i
and impose the commutation relations
[a ,a†] = δ , (44)
i j ij
8