Probabilities and signalling in quantum field theory

0.44 MB·

by  Robert Dickinson

MAN/HEP/2016/01
Probabilities and signalling in quantum field theory
Robert Dickinson,∗ and Jeff Forshaw†
Consortium for Fundamental Physics,
School of Physics and Astronomy, University of Manchester,
Manchester M13 9PL, United Kingdom.
Peter Millington‡
6
1
School of Physics and Astronomy, University of Nottingham,
0
2
Nottingham NG7 2RD, United Kingdom.
r
a
M (Dated: March 29, 2016)

8 We present an approach to computing probabilities in quantum field theory for a
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wideclassofsource–detectormodels. Theapproachworksdirectlywithprobabilities
]
h
t and not with squared matrix elements, and the resulting probabilities can be written
-
p
e in terms of expectation values of nested commutators and anti-commutators. We
h
[ present results that help in the evaluation of these, including an expression for the

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vacuumexpectationvaluesofgeneralnestingsofcommutatorsandanti-commutators
v
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8 in scalar field theory. This approach allows one to see clearly how faster-than-light
7
7 signalling is prevented, because it leads to a diagrammatic expansion in which the
0
. retarded propagator plays a prominent role. We illustrate the formalism using the
1
0
6 simple case of the much-studied Fermi two-atom problem.
1
:
v
i
X
r I. INTRODUCTION
a
Relativistic quantum field theories respect causality and faster-than-light signalling is
forbidden. This well-known fact is a direct consequence of the vanishing of the commutator
(or anti-commutator) of field operators when evaluated at spacelike separations (e.g. see
ref.[1]). Itis, however, lessclearhowfaster-than-lightsignalling(Einsteincausality)emerges
in explicit calculations, where the Feynman propagator is often ubiquitous. In this paper,
∗ [email protected]
† jeff[email protected]
‡ [email protected]
2
we will develop a means to compute probabilities that resolves this matter in a general way,
by highlighting the role of the retarded propagator. The formalism operates at the level of
cross-sections and probabilities rather than at the level of amplitudes.
The archetypal example of a signalling process is the Fermi two-atom problem [2]. Fermi
considered two point-like atoms, A and B, separated by a distance R. At time t = 0, atom
A is prepared in an excited state and atom B is prepared in its ground state. He calculated
the probability that, at a later time T, atom B should be found in its excited state after
absorbing a photon emitted during the spontaneous decay of atom A, which ends up in its
ground state. Fermi believed this probability should be strictly zero for T < R/c, in order to
respect Einstein causality, and he claimed to prove it [2]. However, Fermi was wrong [3], for
he erroneously approximated an integral over positive frequencies by one over both positive
and negative frequencies. The correct result should have been a non-vanishing probability
for the excitation of atom B for T < R/c. The history of the Fermi problem is worth
recapping and we do so in a footnote [4]. The fact that atom B is instantaneously correlated
with atom A is not a problem for Einstein causality, which is restored if one asks instead
for the probability that B is excited at time T with no restriction on the state of atom A
or the electromagnetic field, i.e. if one makes a local measurement on atom B. This is nicely
elucidated in the case of heavy atoms and without the complication of renormalization in
refs. [9, 16]. If one computes the probability that the detector atom is excited at time T,
regardless of the state of the source atom and the electromagnetic field, then the leading
order contributions to the amplitude are illustrated in Figure 1. Graph (b) is Fermi’s and,
by itself, it leads to a contribution that does not vanish for T < R/c. Adding in the other
contributions (graph (c) multiplied by its conjugate and the interference between graphs (a)
and (d)) precisely cancels the causality-violating terms. Note that this relies on the fact that
an atom in its ground state can fluctuate into an excited state with the emission of a field
quantum. This does not violate energy conservation because of Heisenberg’s uncertainty
principle. Although this treatment involves only bare atomic states, it seems to us that
the idea is robust enough to survive renormalization. In this way, superluminal signalling
is prevented in the weak sense proved in refs. [22–24]. A clear statement of weak causality
can be found in ref. [25]. In essence it says that, although atom B may be excited for any
time T > 0, the excitation probability is independent of the state of atom A if T < R/c.
For example, suppose Alice, who is located at the source atom, aims to transmit a bit of
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information to Bob, who is at the detector. To do this, Alice prepares the state of atom A
at time zero. Because atom B can be spontaneously excited for any T > 0, doing this once
will not be enough to transmit the bit of information reliably. Alice will need to repeatedly
prepare the source atom for each bit she wishes to transmit. Bob will then be able to
measure that bit, to a certain statistical precision, by measuring the probability of finding
the detector atom to be excited.
S D S D S D S D
(a) (b) (c) (d)
FIG. 1: The Feynman diagrams corresponding to the amplitudes relevant to the Fermi
problem. We show only those graphs that give rise to contributions that depend upon the
distance between the source and detector atoms. Solid lines denote the source (S) and
detector (D) atoms and the wavy lines are photons. Time runs upwards.
In what follows, we revisit the question of signalling in quantum field theory. Specifically,
we will present a new and quite general way to compute probabilities in the interaction
picture. This approach makes Einstein causality manifest and has the interesting feature
that we do not need to sum explicitly over unobserved emissions.
II. A SIMPLE SOURCE-DETECTOR MODEL
In order to develop the formalism in a familiar context, we start by considering two
point-like atoms, S and D, separated by a distance R, which act as source and detector
of disturbances in a neutral scalar field, φ. In this section, we will present a formalism
that allows one to compute the probability of finding the system to be in some particular
configuration at time t = T given that it was in some other configuration at time t = 0. We
will consider more general source–detector models in Section III.
We begin by considering a closed system represented by a product of the Hilbert spaces
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of the source atom, detector atom and field: H = H S×H D×H φ. For the Hamiltonian,
we take H = H +H , where H = HS+HD+Hφ and H = HSφ+HDφ. The superscripts
0 int 0 0 0 0 int
refer to the spaces in which the operators act (e.g. in the case of HSφ, this is the product
space H S×H φ). In this section, we will only consider interactions between the atoms and
the field. Field self-interactions will be considered in Section III. Under the free part of the
Hamiltonian, H , each atom X ∈ {S,D} has a complete set of bound states {|nX(cid:105)} with
0
eigenvalues given by HX |nX(cid:105) = ωX |nX(cid:105).
0 n
Atoms S and D are assumed to be static and interact with the field at the fixed, spatial
points xS and xD via transition moments µX , which in this toy scalar field example we will
mn
take to be monopole moments. The full interaction-picture Hamiltonian is then
(cid:90)
(cid:88) (cid:88) (cid:16) (cid:17)
H = ωS|nS(cid:105)(cid:104)nS|+ ωD|nD(cid:105)(cid:104)nD|+ d3x 1(∂ φ)2 + 1(∇φ)2 + 1m2φ2 ,
0 n n 2 t 2 2
n n
H (t) = MS(t)φ(xS,t)+MD(t)φ(xD,t), (1)
int
where MX(t) ≡ (cid:80) µX eiωmXnt |mX(cid:105)(cid:104)nX| and ωX = ωX − ωX. We shall assume that
mn mn mn m n
µX = 0 ∀n, i.e. that emission or absorption of a field quantum always results in a transition
nn
up or down in energy. The Fermi problem has also been discussed in the case of two-level
(Unruh-DeWitt) point-like detectors in ref. [26] and, for a discussion of potential causality
issues in general particle-detector models, see ref. [27].
We suppose that the system is initially (t = 0) described by a density matrix ρ and
0
that the measurement outcome is described by an operator E. In general, E is an element
of a Positive-Operator Valued Measure, and it may be written as a sum over products of
hermitian operators:
(cid:88)
E = ES ED E . (2)
(κ) (κ) (κ)
κ
The superscripts S and D denote the Hilbert space in which the operators act and E acts in
the field Hilbert space. We explicitly consider a single product, E = ESEDE, but the gener-
alization to a sum of such operators is straightforward. The probability of the measurement
P
outcome, , is then given by
P = Tr(Eρ ) , (3)
T
ρ ≡ U ρ U† (4)
T T,0 0 T,0
(cid:16) (cid:90) T (cid:17)
and U = Texp 1 dt H (t) . (5)
T,0 i int
0
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Note that the measurement is quite general and not restricted to probing only the state of
the detector atom. We will consider this restricted case in Section IIB.
One of our goals is to determine the sensitivity of the detector to changes in the prepa-
ration of the source. To this end, we will consider an initial mix of two states |i (cid:105) and
p
|i (cid:105):
g
ρ = γ|i (cid:105)(cid:104)i |+(1−γ)|i (cid:105)(cid:104)i | , (6)
0 p p g g
where
|i (cid:105) = |pSgD0φ(cid:105) ≡ |pS(cid:105)⊗|gD(cid:105)⊗|0φ(cid:105) and |i (cid:105) = |gSgD0φ(cid:105) . (7)
p g
The first corresponds to the source atom being in an excited state (labelled by p) and the
detector atom being in its ground state (labelled by g), whilst the second corresponds to
both the source and detector atoms being in their ground states. In both cases, we suppose
that the field is known to have no excitations. Although this is quite a specific initial state,
the results that follow can easily be generalized to other initial states. Moreover, in much
of what follows the choice of initial state is unimportant. We can define the sensitivity of
the detector, σ :
pg
P
d
σ ≡ = P −P , (8)
pg p g
dγ
where
P ≡ (cid:104)i |U† EU |i (cid:105) (9)
p,g p,g T,0 T,0 p,g
is the measurement probability given the state |i (cid:105) at time t = 0. Of course P and P can
p,g p g
also be written as squared matrix elements. However, we do not perform the calculation this
way; instead, asinrefs.[26,28,29], weuseageneralizationoftheBaker-Campbell-Hausdorff
lemma to commute the operator E through the evolution operator, which gives
∞ (cid:90) T
(cid:88)
P = dt dt ...dt Θ (cid:104)i |F |i (cid:105) , (10)
p,g 1 2 j 12...j p,g j p,g
0
j=0
where
F = E,
0
(cid:104) (cid:105)
F = 1 F ,H (t ) , (11)
j i j−1 int j
and Θ ≡ 1 if t > t > t ... and zero otherwise. Using the notation φX ≡ φ(xX,t ) and
ijk... i j k j j
MX ≡ MX(t ), we may write
j j
(cid:104) (cid:105)
F = 1 F , MSφS +MDφD . (12)
j i j−1 j j j j
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We now show how the F operators can be computed to any order j.
j
A. A general commutator expansion
We start from the following identity for any operators AX,BX ∈ H X and Pφ,Qφ ∈ H φ:
(cid:2) (cid:3) (cid:2) (cid:3)(cid:8) (cid:9) (cid:8) (cid:9)(cid:2) (cid:3)
AXPφ,BXQφ ≡ 1 AX,BX Pφ,Qφ + 1 AX,BX Pφ,Qφ , (13)
2 2
in which {AX,BX} ≡ AXBX +BXAX. With F = E, the first commutator is then
0
(cid:2) (cid:3) (cid:8) (cid:9) (cid:8) (cid:9) (cid:2) (cid:3)
F = 1 ES,MS ED E,φS + 1 ES,MS ED E,φS
1 2i 1 1 2i 1 1
(cid:2) (cid:3)(cid:8) (cid:9) (cid:8) (cid:9)(cid:2) (cid:3)
+ 1ES ED,MD E,φD + 1ES ED,MD E,φD . (14)
2i 1 1 2i 1 1
It will be very convenient to define the following sequences of (hermitian) operators:
(cid:2) (cid:3) (cid:8) (cid:9)
EX ≡ 1 EX,MX , EX ≡ EX,MX ,
...k i ... k ...k ... k
¯
(cid:2) (cid:3) (cid:8) (cid:9)
E...X ≡ 1 E...,φX , E...X ≡ E...,φX . (15)
...k i ... k ...k ... k
¯
Note that the indices on these operators are always time-ordered, with the latest time on
the left. Using this notation,
(cid:0) (cid:1)
F = 1 ESEDES +ESEDES +ESEDED +ESEDED . (16)
1 2 1 1 1 1 1 1 1 1
¯ ¯ ¯ ¯
The F can be expressed in a very compact form by introducing an under-circle notation,
j
E E ≡ E E +E E +E E +E E , (17)
kl kl kl kl kl kl kl kl kl kl
◦◦ •• ¯¯ ¯ ¯ ¯ ¯ ¯¯
which denotes a sum over complementary pairs of commutation operations. Exploiting this
notation gives
F = ESEDE ,
0
(cid:0) (cid:1)
F = 1 ESEDES +ESEDED ,
1 2 1 1 1 1
◦ • ◦ •
(cid:0) (cid:1)
F = 1 ES EDESS +ESEDESD +ESEDEDS +ESEDEDD ,
2 4 12 12 1 2 12 2 1 12 12 12
◦◦ •• ◦ ◦ •• ◦ ◦ •• ◦◦ ••
F = 1(cid:0)ES EDESSS +ES EDE(SSD) +ESED E(SDD) +ESED EDDD(cid:1) . (18)
3 8 123 123 (12 3) (123) (1 23) (123) 123 123
◦◦◦ ••• ◦◦ ◦ ••• ◦ ◦◦ ••• ◦◦◦ •••
The indices in parentheses in the last line indicate a summation over those permutations of
the indices that give rise to unique terms, subject to the time indices being ordered within
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each operator. For example, ES ED = ES ED +ES ED +ES ED. The indices on each E...
(12 3) 12 3 13 2 23 1 ...
operator are fixed by those on the corresponding product of EX operators, i.e. the S or D
···
label associated with each numerical index matches that of the associated EX operator, and
...
its underlining state is complementary to the one it has on EX.
...
The general result for F is extremely simple. It is the sum of all distinct products of
n
operators of the form 2−nESEDE... with every index {1,...,n} appearing once on one of
... ... ...
the EX and once on E...:
... ...
n
(cid:88)
F = 2−n ES ED E(S...S D...D) . (19)
n (1...a a+1...n) (1...aa+1...n)
◦···◦ ◦···◦ •···• •···•
a=0
In the above summation, the set i...j is understood to be the empty set if i > j, resulting
in a factor of EX (with no indices).
B. An example: a local measurement
In this section, we shall focus upon the case of a measurement made only on the D atom,
with no restriction on the state of the S atom or the field. Specifically, we compute the
probability of finding the detector atom in an excited state, |qD(cid:105), at time t = T. In this case
(cid:88)
E = |nS qD αφ(cid:105)(cid:104)nS qD αφ| = IS|qD(cid:105)(cid:104)qD| Iφ , (20)
n,α
where IS and Iφ are the identity operators in H S and H φ. Notice that we have used the
completeness of states to sum over the final states of the source atom and field in Eq. (20).
In this way, we avoid ever having to sum explicitly over unobserved final states. This feature
of our approach may have interesting consequences for calculations of inclusive observables
in S-matrix theory, where, for example, the sum over unobserved emissions is important in
securing the cancellation of infra-red singularities in gauge theories.
Since we fix E = Iφ, it follows that
EX... = 0 , EX = 2φX ,
1... 1 1
¯
EXY = 2[φX,φY] , EXY = 2{φX,φY} ,
12 i 1 2 12 1 2
¯ ¯¯
EXYZ... = 0 . (21)
123...
¯
The first of these relations immediately sets half of the terms in Eq. (19) to zero and ensures
that the ‘1’ index (which labels the latest time) is never underlined on an EX operator.
1...
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The first and last of the relations in Eq. (21) are examples of a more general rule: any
E... operator vanishes if its first k indices consist of more non-underlined than underlined
...
indices, for any k. In Appendix A, we show how to evaluate any E... operator and its vacuum
...
expectation value, given E = Iφ.
Since we also fix ES = IS (we will consider the case of non-trivial ES in the next sub-
section), it further follows that
ES = 0 , ES = 2MS . (22)
k... k k
¯
The first of these eliminates half of the remaining terms in Eq. (19) and ensures that the first
index in ES is always underlined. The ‘1’ index must now be carried by the ED operator.
k... 1...
¯
With these restrictions, up to fourth order, the non-vanishing terms in Eq. (19) are
F = 1EDED ,
1 2 1 1
¯
(cid:0) (cid:1)
F = 1 EDEDD +EDEDD +EDESEDS ,
2 4 12 12 12 12 1 2 12
¯¯ ¯ ¯ ¯ ¯
(cid:0) (cid:1)
F = 1 ED EDDD +ED EDDD +EDESEDDS +EDESEDSD +EDES EDSS ,
3 8 123 123 123 123 12 3 123 13 2 123 1 23 123
◦ ¯¯• ¯ ¯ ¯ ¯ ¯¯ ¯ ¯ ¯ ¯ ¯ ¯
(cid:0)
F = 1 ED EDDDD +ED EDDDD +ED ESEDDDS +ED ESEDDDS
4 16 1234 1234 1234 1234 123 4 1234 123 4 1234
◦◦ ¯¯•• ¯ ◦ ¯ ¯• ◦ ¯ ¯¯• ¯ ¯ ¯ ¯
+ED ESEDDSD +ED ESEDSDD +EDES EDDSS +EDES EDSDS
124 3 1234 134 2 1234 12 34 1234 13 24 1234
◦ ¯ ¯¯ • ◦ ¯ ¯ ¯• ¯◦ ¯¯ • ¯◦ ¯ ¯•
(cid:1)
+EDES EDSSD +EDES EDSSS . (23)
14 23 1234 1 234 1234
◦ ¯ ¯ ¯• ¯ ◦ ¯ ¯•
To compute the measurement probability, we need the expectation values of these operators.
(This is where the dependence upon the initial state of the system enters.) In Appendix B,
we present rules to evaluate the expectation values of general atom operators, EX. For what
···
follows in this section, the following expectation values are useful
(cid:104)gD|ED|gD(cid:105) = 0 ,
1
(cid:104)gD|ED |gD(cid:105) = |µD|22sinωDt , (cid:104)gD|ED |gD(cid:105) = |µD|22cosωDt ,
12 qg qg 12 12 qg qg 12
¯
(cid:104)pS|ES|pS(cid:105) = 2µS ,
2 pp
¯
(cid:88) (cid:88)
(cid:104)pS|ES |pS(cid:105) = |µS |24cosωS t , (cid:104)pS|ES |pS(cid:105) = |µS |24sinωS t , (24)
23 pn pn 23 23 pn pn 23
¯¯ ¯
n n
where t ≡ t −t . For a non-zero contribution to the signal sensitivity σ with µ = 0,
ij i j pg nn
we require (cid:104)gD| ED |gD(cid:105) =(cid:54) 0 and (cid:104)pS| ES |pS(cid:105) =(cid:54) (cid:104)gS| ES |gS(cid:105), so need only keep terms with
... ... ...
at least two indices on each of ED and ES, which means that the first non-zero contribution
... ...
to σ arises at fourth order.
pg
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P
Note that the leading-order contribution to the transition probability actually comes
p
from F :
2
(cid:16) (cid:17)
(cid:104)i |F |i (cid:105) = (cid:104)pSgD0φ| 1 EDEDD +EDEDD +EDESEDS |pSgD0φ(cid:105)
p 2 p 4 12 12 12 12 1 2 12
¯¯ ¯ ¯ ¯ ¯
(cid:16) (cid:17)
= |µD|2 ∆DD(H)cosωDt + ∆DD(R)sinωDt . (25)
qg 12 qg 12 12 qg 12
However, this does not depend on the state of the source atom and cancels when computing
σ . For initial states other than the ones we consider in this section, (cid:104)F (cid:105) can contribute
pg 2
to σ . This occurs when the initial density operator ρ contains states that are oblique
pg 0
with respect to the projection operator ED, as would be the case if we replaced |gD(cid:105) with
(cid:0) (cid:1)
√1 |gD(cid:105) + |qD(cid:105) in |i (cid:105) and |i (cid:105). Additionally, (cid:104)F (cid:105) contributes to σ when ρ contains
p g 3 pg 0
2
superpositions of field states differing by a single field quantum.
Returning to the calculation of σ , using Eqs. (24) and (A10), the contributing terms
pg
are
(cid:16) (cid:17)
(cid:104)i |F |i (cid:105) ⊃ (cid:104)pSgD0φ| 1 EDES EDDSS +EDES EDSDS +EDES EDSSD |pSgD0φ(cid:105)
p 4 p 16 12 34 1234 13 24 1234 14 23 1234
¯◦ ¯¯ • ¯◦ ¯ ¯• ◦ ¯ ¯ ¯•
(cid:16) (cid:17)
= 1 (cid:104)ED(cid:105) (cid:104)ES (cid:105)(cid:104)EDDSS(cid:105)+(cid:104)ES (cid:105)(cid:104)EDDSS(cid:105)
16 12 34 1234 34 1234
¯ ¯¯ ¯ ¯¯ ¯¯
(cid:16) (cid:17)
+ 1 (cid:104)ED(cid:105) (cid:104)ES (cid:105)(cid:104)EDSDS(cid:105)+(cid:104)ES (cid:105)(cid:104)EDSDS(cid:105)
16 13 24 1234 24 1234
¯ ¯ ¯¯ ¯¯ ¯ ¯
+ 1 (cid:104)ED(cid:105)(cid:104)ES (cid:105)(cid:104)EDSSD(cid:105)+ 1 (cid:104)ED(cid:105)(cid:104)ES (cid:105)(cid:104)EDSSD(cid:105) (26)
16 14 23 1234 16 14 23 1234
¯ ¯ ¯¯ ¯ ¯ ¯ ¯
(cid:88) (cid:110) (cid:16) (cid:17)
= 2 |µS |2|µD|2 cosωDt sinωS t ∆DS(H) +cosωS t ∆DS(R) ∆DS(R)
pn qg qg 12 pn 34 24 pn 34 24 13
n
(cid:16) (cid:17)
+ cosωDt sinωS t ∆DS(H) +cosωS t ∆DS(R) ∆DS(R)
qg 12 pn 34 14 pn 34 14 23
(cid:16) (cid:17)
+cosωDt sinωS t ∆DS(H) +cosωS t ∆DS(R) ∆DS(R)
qg 13 pn 24 34 pn 24 34 12
(cid:16) (cid:17) (cid:111)
+ sinωS t cosωDt ∆SD(H) +sinωDt ∆SD(R) ∆DS(R) . (27)
pn 23 qg 14 34 qg 14 34 12
XY(R) XY(H)
Theretarded(∆ )andHadamard(∆ )fieldpropagatorsaredefinedinAppendixA.
ij ij
Wemayrepresentanytermin(cid:104)F (cid:105)graphically, forarbitaryevenn, usingthefollowingrules:
n
1. Draw two lines moving forwards in time, corresponding to S and D.
2. Draw n vertices associated with the times t (latest) to t (earliest) and distribute
1 n
them between the two lines with the latest time vertex residing on the D line.
10
3. If there are vertices on S, draw a propagator line between the latest vertex on S and
a later vertex on D. Pair all other vertices in any combination and join each pair with
a propagator line.
4. For every vertex at the earlier end of a propagator line, either do nothing or circle the
vertex and draw an arrow on the associated propagator. The exception is the latest
vertex on the S line, which is always circled and its associated propagator is always
arrowed.
5. Associate a factor (cid:104)EX (cid:105) with each line X, where {i,j,...} is the set of vertices on X,
ij...
and underline every index corresponding to a circled vertex.
XY(R)
6. Associate a factor ∆ with each arrowed propagator line from t on X to t on Y
ij i j
XY(H)
and a factor ∆ with each non-arrowed propagator line.
ij
7. Write a factor (1)n2−1, with a further factor of 1 if all vertices reside on D.
2 2
The relations underlying these rules are derived in Appendix A, and we show the 8 graphs
contributingto(cid:104)F4(cid:105)inFigure2. Ingeneral,thenumberofgraphsin(cid:104)Fn(cid:105)is2n2−1(n−3)!!(2n+
n−3) = {3, 34, 804, 31320,...} for n = {2,4,6,8,...}, of which 2n2−1(n−3)!!4(n−1)(cid:0)n−1(cid:1) =
n+2 n/2
{1, 8, 180, 6720,...} have an equal number of vertices on each line, i.e. no loops ∆XX.
ij
DS(R)
Crucially, every term in Eq. (27) contains a retarded propagator ∆ with 0 < t <
ij j
t < T, implying that every term in σ vanishes to fourth order when T < R, where
i pg
R ≡ |xD−xS|. The stated rules ensure that this holds to all orders. This is in accord with
the demands of Einstein causality, i.e. observation of the detector atom is insensitive to the
state of the source atom for times T < R.
In the next section, we will verify that σ vanishes for spacelike separations to all orders
pg
for more general source–detector systems. But, before moving away from the two-atom
problem, we will consider the probability of finding, at the time T, the detector atom in
an excited state (qD) and the source atom in its ground state (gS). Again we will make no
restriction on the state of the field at this time. Because this involves measuring the state of
two atoms at the same time, it is not local and the probability need not vanish for T < R.