Violation of the strong Huygen's principle and timelike signals from the early Universe

0.41 MB·English

by  Ana Blasco

Violation of the strong Huygen’s principle and timelike signals from the early Universe
Ana Blasco,1 Luis J. Garay,1,2 Mercedes Mart´ın-Benito,3 and Eduardo Mart´ın-Mart´ınez4,5,6
1Departamento de F´ısica Te´orica II, Universidad Complutense de Madrid, 28040 Madrid, Spain
2Instituto de Estructura de la Materia (IEM-CSIC), Serrano 121, 28006 Madrid, Spain
3Radboud University Nijmegen, Institute for Mathematics, Astrophysics and Particle Physics,
Heyendaalseweg 135, NL-6525 AJ Nijmegen, The Netherlands
4Institute for Quantum Computing, University of Waterloo, Waterloo, Ontario, N2L 3G1, Canada
5Department of Applied Mathematics, University of Waterloo, Waterloo, Ontario, N2L 3G1, Canada
6Perimeter Institute for Theoretical Physics, Waterloo, Ontario, N2L 6B9, Canada
WeanalyzetheimplicationsoftheviolationsofthestrongHuygensprincipleinthetransmission
of information from the early universe to the current era via massless fields. We show that much
more information reaches us through timelike channels (not mediated by real photons) than it is
5 carried by rays of light, which are usually regarded as the only carriers of information.
1
0
2 Introduction.— Quantum field theory in curved a similar way as it was anticipated in [5]. This leads

r spacetimesprovidesthenaturalframeworktoinvestigate us to the following conclusion: Signals received today
p
the quantum nature of matter in the presence of grav- willgenericallycontainoverlappedinformationaboutthe
A
ity. In the context of relativistic quantum information, past from both timelike and light connected events.

2 quantum field entanglement can be used as a powerful Furthermore,theinformationpropagatinginthetime-
2 physical resource in the analysis of several phenomena likezonedecaysslowerthanwhatonewouldexpectasthe

such as detection of spacetime curvature or the trans- spatialdistancebetweensenderandreceiverincreases,as
]
h mission of information in relativistic settings [1–5]. opposedtotheinformationcarriedbylight. Remarkably,
p Quantum entanglement might play an important role we will see that in a matter-dominated universe, the in-
- inthestudyoftheearlyuniverse(forareview,see[6,7]). formation propagating in the interior of the light cone
t
n For example, the phenomena known as entanglement doesnotdecayatallwiththesender-receiverspatialdis-
a
‘harvesting’ and ‘farming’ (i.e. swapping of entangle- tance.
u
q ment from a quantum field to particle detectors [1, 8]) Set-up.— We will consider a spatially flat and open
[ is strongly influenced by the cosmological background as Friedmann-Robertson-Walker (FRW) spacetime:

2 proven by Ver Steeg and Menicucci [2]. ds2 =a(η)2( dη2+dr2+r2dΩ2). (1)
v Another interesting result comes from the conse- −
0 quencesinrelativisticquantumcommunicationofthevi- Natural units (cid:126) = c = 1 are used throughout. This
5 olations of the strong Huygens principle [5]. This princi- geometry is generated by a perfect fluid with a con-
6 plestatesthattheradiationGreen’sfunctionhassupport stant pressure-to-density ratio p/ρ = w > 1, so that
1 onlyonthelightcone[9–12]. Asaconsequence, commu- the scale factor evolves as a ηα+1/2 t(2−α+1)/(2α+3),
0 ∝ ∝
. nication through massless fields is confined to the light with α=(3−3w)/(6w+2)>−3/2. We note that for
1 cone. This is true in four-dimensional flat spacetime but all w > 1, these cosmologies display a Big Bang singu-
0 −
not in the presence of curvature. These violations have larity. Heretisthecomovingtimedt=a(η)dη. Forcom-
5
1 been studied before in the context of cosmology for clas- putationalsimplicity,inthisbackgroundwewillconsider
: sical fields [13, 14]. a test massless scalar field Φ quantized in the adiabatic
v
The violation of the strong Huygens principle implies vacuum[15]. Noticethatalthoughtheadiabaticvacuum
i
X that there can be a leakage of information towards the isinitselfaninterestingobjectofstudy,thechoiceofthe
r insideofthelightcone,evenformasslessquantumfields. field’s state is not relevant for our results as pointed out
a
Whenthishappens,itispossibletobroadcastamessage below. Wealsointroduceacoupleofcomovingobservers
to arbitrarily many receivers with the energy cost being AliceandBob. Theycanperformindirectmeasurements
spent by the receivers of the message [5]. Here we will on the field by locally coupling particle detectors.
argue that the violation of the strong Huygens principle For arbitrary detector trajectories, the interaction be-
has unexpected consequences in cosmological scenarios, tween the field and the particle detectors will be de-
in particular in the propagation of information from the scribed by the Unruh-DeWitt model [16], which displays
early universe to the current era. all the fundamental features of the light-matter interac-
tionwhenthereisnoexchangeoforbitalangularmomen-
We will study conformal and minimal couplings of a
tum[17,18]. ThecorrespondinginteractionHamiltonian
test massless scalar field in a cosmological background.
(in the interaction picture) for each detector is given by
Wewillshowthatwhiletheconformalcasedoesnotallow
(cid:90)
for the leakage of information into the future light cone,
H =λ χ (t)µ (t) d3xa(t)3F[x x (t),t]Φ[x,η(t)],
the minimal coupling generically allows for information- I,ν ν ν ν − ν
carrying violations of the strong Huygens principle, in (2)
2
twehcteorer.ν =0 {≤A,Bχν}(td)en≤ote1s eiisthtehreAsliwceit’cshoinrgBfoubn’sctidoen- GαJY(η,η(cid:48),k)= Yα(kη(cid:48))LJαJ(αk(ηk(cid:48)η))−YαJ(αk(ηk(cid:48)η)(cid:48))LYα(kη(cid:48)),
oµfν(tth)e=deteeνctogrν eνiΩ,νtλν+isgνitseνceo−uipΩlνintgisstitrsenmgtohn,opaonlde LJα(kη)=Jα−1(kη)−Jα+1(kη), (5)
| (cid:105)(cid:104) | | (cid:105)(cid:104) |
moment (where |gν(cid:105) and |eν(cid:105) are its ground and ex- andGαYJ andLYα aredefinedanalogouslyexchangingthe
cited states, and Ων is its energy gap). xν(t) is the Bessel functions Jα and Yα.
detector’s trajectory, which for the comoving case be- The case of a cold-matter-dominated universe, for
comes xν = const. The field operator Φ is evaluated which α = 3/2, is of particular interest due to its sim-
along the worldline of the comoving detectors, which plicity. In this case the commutator reduces to [19]
are spatially smeared according to a Gaussian distribu-
(cid:34)
tion F(x,t)=(σ√π)−3e−a(t)2x2/σ2, where σ character- (cid:2)φ(x,t),φ(x(cid:48),t(cid:48))(cid:3)= i δ(∆η+R)−δ(∆η−R)
izes the constant physical size of the detector. This pro- 4π a(t)a(t(cid:48))R
file also regularizes the UV divergences that appear in (cid:35)
the case of point like detectors [18]. θ( ∆η R) θ(∆η R)
+ − − − − . (6)
Signalling.— In order to study whether Alice and a(t)a(t(cid:48))η(t)η(t(cid:48))
Bob will be able to communicate through the field,
we analyze the signalling estimator introduced in [5], We explicitly see the violation of the strong Huygens
which determines how the excitation probability of B is principle: The commutator gives a non-vanishing contri-
modulated by the interaction of A with the field. Let bution to the signalling estimator even when the events
ψ =α e +β g be the initial state of the detec- (x,t) and (x(cid:48),t(cid:48)) are timelike separated, due to the
0,ν ν ν ν ν
|tor ν(cid:105). At le|ad(cid:105)ing or|der(cid:105)in time-dependent perturbation θ-term. Let us note that the δ-term is the commutator
theory,thisestimatorreadsS =λ λ S + (λ4),where oftheconformallycoupledmasslessscalarfielddiscussed
A B 2 O ν above, and it decays as the comoving distance R grows.
(cid:90) (cid:90)
S =4 dv dv(cid:48)χ (t)χ (t(cid:48))Re(α∗β eiΩAt)F(x x ,t) Notice that, on the other hand, the contribution of the
2 A B A A − A commutatorinsidethelightcone(θ-term)doesnotdecay
F(x(cid:48) x ,t(cid:48))Re(cid:16)α∗β eiΩBt(cid:48)(cid:2)φ(x,t),φ(x(cid:48),t(cid:48))(cid:3)(cid:17), (3) asRincreases. Thiswillhaveimportantconsequencesin
× − B B B the transmission of information from A to B.
and dv = a(t)3d3xdt. This expression generalizes the Pointlike detectors.— The probability of excitation
of a sharply switched pointlike detector is UV divergent
corresponding expression in [5] to our case of smeared
[20]. However, from the commutator (6) we see that
detectors, and as it can be seen in [4], it is independent
the signalling estimator (3) is UV-safe in the pointlike
of the initial state of the field. Let us study the form of
detector limit (σ 0), even considering sharp switch-
the field commutator in the cosmological spacetime that
→
ing. Hence, since the pointlike limit is distributionally
we are considering, both for conformally and minimally
well behaved, one can take the abrupt switching func-
coupled massless scalar fields.
tion χ (t) = 1 if t [T ,T ] and zero otherwise. The
Forconformalcoupling,fieldmodesaregiven—except ν iν fν
∈
result is finite and given by
for an overall 1/a(t) factor— by plane waves in confor-
maltimeη. Therefore, thecommutatoristhesameasin 1
Minkowski spacetime, except for overall conformal fac- S2 = πRe(αA∗βA)Im(αB∗βB)[Sδ+Sθ], (7)
tors, and vanishes if the events (x,t) and (x(cid:48),t(cid:48)) are not
where S and S are respectively the contributions to
lightconnected. Hence,thereisnoviolationofthestrong δ θ
(3) coming from the Dirac delta and the Heaviside theta
Huygens principle [10]: Communication is only possible
terms in (6). They can be written in terms of polyloga-
strictly on the light cone.
rithmic functions Li (z) for the different cases shown in
In contrast, for minimal coupling, the above commu- s
Fig. 1 and detailed in Table I. Using the short notation
tator does not generically have support only on the light
η η(T ), η η(T ), their explicit expressions are
cone. For the cosmological spacetime (1), iν iν fν fν
≡ ≡
(cid:0) (cid:1)
(cid:2)φ(x,t),φ(x(cid:48),t(cid:48))(cid:3)=iθ(−π2∆aη(t))−a(tθ(cid:48)()∆Rη) (4) Sδ =(cid:40)(zl1n−(cid:16)ηzf2A)θ(cid:17)zln1(cid:16)−ηfzB2(cid:17),, case 5 (Table I()8,)
(cid:90) ∞dksin(kR)g (cid:0)η(t),η(t(cid:48)),k(cid:1), Sθ = [L(zηi)A L(z η)iB+N(z )]θ(z z ), other cases,
α 1 2 1 1 2
× − −
0
where, R = x x(cid:48) , ∆η = η(t) η(t(cid:48)), and g can be where we define
α
expressed as(cid:107)rat−iona(cid:107)l functions of−first and second kind (cid:18)R(z 1)(cid:19)
L(z)=ln − ln(z)+Li (1 z), (9)
Bessel functions as follows: η 2 −
iA
(cid:114)η(cid:104) (cid:105) (cid:18)R(z 1)(cid:19) (cid:16)η (cid:17)
gα(η,η(cid:48),k)= η(cid:48) GαJY(η,η(cid:48),k)+GαYJ(η,η(cid:48),k) , N(z)=ln η− ln RfzB , (10)
iA
3
6
η 44
TIMELIKE
à
→
5 3 5 4 3 2 1
ηfB 4 3 2 2)B3 → à
ηiB Bob 1 6 1.5 2λλA22 aæ æ à æCδ
T LIK E 1.0 510− æ æ æ æ à à àC
ηfA LIG H 0.5 C( 11 à à à à à à àæ à à à æ æ æ æ æà
Alice SPACELIKE æ à
η 0.0 æà æà æà00æàæ æ æ æ æ æ æ æ ææææ æ æ æ æà æ æà æà æàæà æà æà æà
iA -xxx 1000 2000 1310000 40002205000 6033000 7000 440
R R (∆/30)
FIG. 1. Different causal relationships between Alice and 22..55 à à
Bob’s detectors switching periods. These cases are explicitly 1 2 à à à 3 → 4 5
specified in Table I. Recall that ηiν ≡η(Tiν), ηfν ≡η(Tfν). 22..00 → à à
) æ à
2B à
1.5 2λA11..55 æ æ
min(η +R,η ) max(η +R,η ) λ bæ æ æCδ
z1 = fAR fB , z2 = iAR iB . 1.0 50− 11..00 æ æ æ æ à àC
1 æ æ
(11)0.5 C( 00..55 æ æ æ
For simplicity, in Eqs. (7)-(8) we have already par-0.0 æà æà æà00æà..00æà æà æà æà æ æ æà æ ææææàææàææàæàæàæàæàæàæà æà æà æà æà
ticularized the study to the case of zero-gap detectors, 1000 200100100300200200400300300500400400600500500700600600 70700
Ων = 0. This choice is arbitrary and has no effect on TiB (10∆/3)
our main results. Moreover, it is not uncommon to find
relevantatomictransitionsbetweendegenerate(orquasi-
FIG. 2. Channel capacity (in bits) and its δ-term as func-
degenerate) atomic energy levels, for example, atomic
tionsof(a)thespatialseparationbetweenAliceandBob,for
electron spin-flip transitions. T −T =T −T =∆, T = ∆/30, and T = 10∆,
fA iA fB iB iA iB
Channel capacity.— Let us now compute the capac- (b) the temporal separation between Alice and Bob. In (b),
we vary T while keeping T − T = T − T = ∆
ity of a communication channel between an early Uni- iB fA iA fB iB
constant and we fix T = ∆/30 and R = ∆/10. Different
verse observer, Alice, and a late-time observer, Bob. To iA
regions are labelled according to the case numbers of Fig. 1
obtain a lower bound to the capacity, we use a simple
and Table I. Since both detectors remain switched on during
communication protocol: Alice encodes “1” by coupling thesameamountofpropertime,onlycases1to5occur. The
her detector A to the field, and “0” by not coupling it. violation of strong Huygens can be seen in region 5 (timelike
Later, Bob switches on his detector B and measures its separation).
state. If B is excited, Bob interprets a “1”, and a “0”
otherwise. Thecapacity ofthisbinaryasymmetricchan-
given, at leading order, by
nel (i.e., the number of bits per use of the channel that
AlicetransmitstoBobwiththisprotocol)wasprovento
(cid:18) (cid:19)2
2 S
be non-zero [5], no matter the level of noise, and it is C λ2λ2 2 + (λ6). (12)
(cid:39) A Bln2 4α β O ν
B B
| || |
Figures 2a and 2b show the behavior of the channel ca-
pacity C. For comparison, we also display the channel
TABLE I. Cases of causal relationships. See Fig. 1.
capacity in the conformally coupled case, C . We have
δ
Case Conditions selected initial detector states that, in our case, maxi-
1 ηfB ≤ηiA+R mize the channel capacity (i.e. αA = βA = 1/√2,
2 ηiB <ηiA+R<ηfB ≤ηfA+R arg(αA) arg(βA)=π, arg(αB) | arg|(βB|)=|π/2).
− −
3 η ≥η +R, η ≤η +R Let us first analyze how the ability of Alice to signal
iB iA fB fA
4 η >η +R>η ≥η +R Bob depends on their time separation. From the δ-term
fB fA iB iA
5 η ≥η +R of Eq. (6) we see that the information transmitted by
iB fA
‘raysoflight’decayswiththedistancebetweenAandB,
6 η <η +R, η >η +R
iB iA fB fA
becomingnegligibleforlongtimes. Thisyieldstheunsur-
4
prisingresultthatthecapacityofthelight-likecommuni- geneous and isotropic spacetimes, and obtained fully-
cation channel between the early universe and nowadays analytic closed expressions for the channel capacity.
becomes negligible. In fact the channel capacity decays More importantly, we have shown that the transmis-
essentially with the square of the distance between Alice sion of information via timelike violations of the strong
and Bob. Notice that the information carried by ‘rays Huygens principle decays more slowly than the informa-
of light’ constitutes the only contribution to the channel tion carried by ‘rays of light’. For the particular case of
capacity in the conformally coupled case, where strong minimalcouplingandacold-matter-dominateduniverse,
Huygensprincipleisfulfilled(seeCδ inFigs. 2aand2b). the channel capacity between timelike separated sender
Very remarkably, the θ-term in (6) does not explicitly and receiver does not decay at all with their spatial sep-
decay with the distance between A and B. Instead, it aration. We have also shown that the temporal (loga-
is inversely proportional to the conformal time between rithmic) decay in the amount of information that can be
the Big Bang and both A and B. Of course, this means transmitted through the ‘Huygens channel’ can be com-
thattherewillbealate-timedecayinthechannelcapac- pensatedbydeployinganetworkofreceiversspreadover
ity (see Figs. 2a and 2b). However, this decay can be the interior of the future light cone of the sender.
(over)compensated by deploying a number of spacelike
Although we studied the simpler case of a scalar field,
separated B receivers, which fill the interior of Alice’s
the strong Huygens principle is violated for the electro-
light cone in a given time slice. This does not entail in-
magnetic field A as well [21]. Interestingly, the electro-
µ
creasing the information gathered by every B receiver,
magnetictensorF isconformallyinvariant,andthusit
µν
butinsteadimpliesthateveryB couldberegardedasan
doesnotdisplaystrongHuygensprincipleviolations[22].
approximatelyindependentuserofthechannelthatcould
However, sincethecouplingofthechargedcurrentswith
combine their statistics with the other receivers later on.
the electromagnetic field is through A , electromagnetic
µ
Notice that there would be some entanglement harvest-
antennaswillseethestrongHuygensprincipleviolations
ing between these spacelike separated B receivers [1, 2],
in the same fashion as they see e.g. the Aharonov-Bohm
whichwouldinturncorrelatetheiroutcomestosomeex-
effect or Casimir forces. Even simple protocols, as the
tent. Nevertheless, these harvesting correlations can be
one studied here, show that a considerable amount of
made small (e.g. turning down λ while keeping λ λ
B A B information is encoded in the Casimir-like interactions
constant) so that the B’s become approximately inde-
(not mediated by real photons [4, 5]) between timelike
pendent users of the channel. This (over)compensation
separated events.
is possible because the number of receivers that are in
In summary, we conclude that all events that usually
timelike contact with A increases as the volume of the
generate light-signals also generate timelike signals not
light cone (proportional to η) while the signalling term
mediated by photon exchange, which may in fact carry
decays only logarithmically, as we can see in case 5 of S
θ
more information than light-like signals. In particular,
in Eq. (8) as well as in region 5 of Fig. 2b.
inflationary phenomena, early universe physics, primor-
Conclusions.— We have studied how the violations
dial decouplings, etc., will leave a timelike echo that de-
of the strong Huygens principle in quantum communica-
cays slower than the information carried by light. This
tion, proposed in [5], allow for the transmission of infor-
might allow us in principle to obtain more information
mationfromtheearlystagesoftheUniversetonowadays.
about the early universe than simply observing the elec-
We have focused on a simple lower bound to the ca-
tromagnetic radiation.
pacityofacommunicationchannelbetweenanearlyUni-
verse observer and a late-time observer who use Unruh- Acknowledgments.— The authors would like to give
DeWitt particle detectors to transmit and receive infor- many thanks to Achim Kempf and Jorma Luoko for our
mation through their local interaction with a massless very helpful discussions and their valuable insights. The
quantum field. authors would also like to thank Robert Jonsson for our
We have seen, on very general grounds, that the vio- fruitful discussions. L.J.G. and M.M-B. acknowledge
lation of the strong Huygens principle enables the trans- financial support from the Spanish MICINN/MINECO
mission of information between events that are timelike Project No. FIS2011-30145-C03-02 and its continuation
separated. This is so even though the receiver cannot No. FIS2014-54800- C2-2-P. M. M-B. M.M-B. also ac-
receive real quanta from the sender. This is a very gen- knowledgesfinancialsupportfromtheNetherlandsOrga-
eral phenomenon in cosmological backgrounds, the most nization for Scientific Research (Project No. 62001772).
notableexceptionsbeingmasslessfieldsconformallycou-
pled or minimally coupled to radiation-dominated uni-
verses (α = 1/2). The cases of universes dominated
by perfect fluids for which the strong Huygens princi-
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