Advanced Studies in Theoretical Physics
Vol. 11, 2017, no. 1, 19 - 26
HIKARI Ltd, www.m-hikari.com
https://doi.org/10.12988/astp.2017.6929
Graviton Decay via Excitation
of Nonlinear Vacuum Fluctuations
Giovanni Modanese
Free University of Bolzano, Faculty of Science and Technology
P.za Università 5, Bolzano, Italy
c 2016 Giovanni Modanese. This article is distributed under the Creative
Copyright Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
Abstract
Gravitons relate to gravitational waves as photons to electromagnetic waves, but with the important difference that Einstein gravity is
intrinsically nonlinear. This could lead in principle to the decay of a
graviton into many collinear gravitons of smaller frequency, but such a
process is found to have a very small probability. Another possibility is
the decay of gravitons by interaction with a class of vacuum fluctuations
made of weakly-bound couples of virtual masses. In this case the final
state is a continuum of excited and Lorentz-boosted bound states. We
estimate the decay probability by evaluating the relevant matrix elements and phase space volume, with a suitable cut-off. The calculation
also leads to further insights into the properties of these vacuum fluctuations. We find that the probability of this decay process is also very
small, and compatible with the existence of gravitational waves with
very long propagation range.
Subject Classification: 04.20.-q Classical general relativity. 04.60.-m
Quantum gravity
Keywords: Gravitational waves; gravitons; vacuum fluctuations
1
Introduction
The first direct observation of gravitational waves [1] has renewed the interest
into effective field theories of gravity based on the Einstein action [2, 3] and
20
Giovanni Modanese
into the connection between classical wave solutions and quantum states, or
gravitons. Since the connection between electromagnetic waves and photons
plays a crucial role in QED and actually in quantum mechanics at large, it is
fair to expect that a full understanding of gravitons and their properties is an
important requirement for successful quantum gravity models [4, 5]. In previous work [6, 7] we have analysed the possible decay of a graviton into gravitons
of smaller frequency, in a typical non-linear process peculiar of gravitational
interactions. We have found that in the absence of a cosmological constant
Λ the probability of this process in Einstein quantum gravity is exactly zero,
for reasons of gauge symmetry (Ward identities). We also found that due to
Lorentz invariance, even in the presence of a cosmological constant the decay
probability must have in natural units the general form
∞
1 X
dP
= τ −1 =
cj (ΛG)j
dt
EG j=1
(1)
where cj are adimensional constants. Restoring the explicit dependence on
c and h̄ one finds that with j = 0 the decay probability is not proportional
to h̄; this term would then describe a classical “frequency degradation” of
gravitational waves, a phenomenon which is ruled out by the recent observations. With j > 0, considering that ΛG is of the order of 10−120 [8], the decay
probability is negligible.
In this work we consider a process in which a graviton decays by exciting
a pair of nonlinear vacuum fluctuations peculiar of the Einstein action [9, 10].
The computation of the probability of this process is an important test for
a theory of effective quantum gravity which takes vacuum fluctuations into
account. Specifically, it is important
(1) For the internal consistence of the model, because the process is peculiar and complex, involving a continuum of excited states, which in addition
are Lorentz-invariant, so that a double integration on energy and on a boost
parameter β is required.
(2) For external consistence, because it is expected that the final result for
the decay probability is finite and sufficiently small to be compatible with the
observed propagation capability of gravitational waves.
Both tests are passed, as we show in the following. In the derivation we also
obtain new insights on the nature and properties of the vacuum fluctuations
involved.
21
Graviton decay
2
Matrix elements
The linearized Hamiltonian coupling between a gravitational wave and n particles is in general [12, 13]
Hint =
n
X
1 i j
pa pa hij (t, x)
a=1 2ma
(2)
where ma , pa are the mass and momentum of the a-particle and hij (t, x) is the
metric field of the wave. This formula can be used, for instance, to compute
the probability that the wave excites an atomic wavefunction. More generally,
one is interested into a matrix element between an inital state Ψin and a final
state Ψf in , of the form
n
X
hΨf in |
a=1
1 i j
p p |Ψin i hij (t, x)
2ma a a
(3)
α
The metric of the plane wave is hµν ∝ uµν eik xα , being uµν a polarization
tensor which in the TT gauge has as non-vanishing components for propagation
in the z direction u11 = 1, u22 = −1, u12 = u21 = i.
More precisely, we have
hµν (t, x) =
4πGh̄F
kc4
!1
2
uµν eik
αx
α
+ c.c.
(4)
with k = 2π/λ and F graviton flux (number of gravitons per unit area and
time).
As shown in [9, 10] in the path integral formalism, the vacuum state of
pure Einstein quantum gravity contains a continuum of vacuum fluctuations
which take the form of the fields of negative virtual masses. (See also a related
equipotential lines technique in [11].) We call such field configurations “zeromodes”, because their gravitational action is zero (but not minimum). These
virtual masses interact, and the pair interaction gives rise to a continuum
of excited states. We have computed the decay probability of these excited
states and the probability that they are excited by coherent or incoherent
matter. Results to leading order are consistent so far, thus giving a model of
effective quantum gravity which takes into account non-perturbative vacuum
fluctuations. In this context, it is interesting to see how a pair of identical
localized zero-modes |1i, |2i react to external gravitational waves, i.e., what
is their excitation probability by the gravitons of the wave, and consequently
the graviton lifetime.
We take as initial and final state Ψin , Ψf in , respectively, the symmetrical
ground state Ψ+ = √12 (|1i + |2i) of the zero-mode pair and its first antisymmetrical excited state Ψ− = √12 (|1i − |2i). By expanding (3) we find the
22
Giovanni Modanese
Figure 1: Pair of identical localized zero-modes at distance r.
expression
hΨf in |Hint | Ψin i =
h
i
1 X
(h1| + h2|) h11 (pxa )2 + h22 (pya )2 (|1i + |2i)
2 a=1,2
(5)
The index a refers to the virtual particles |1i, |2i; the operator pa can give a
non-zero result only when it acts on its own particle.
In (5) we have omitted the terms proportional to h12 : these vanish because
the momenta along x and y are uncorrelated. The terms in (5) where (pxa )2
and (pya )2 act between h1| and |1i or between h2| and |2i do not vanish, due
to the uncertainties ∆px and ∆py (the states |1i and |2i are quite localized);
since, however, h11 = −h22 , these terms cancel each other. Finally, consider
the remaining terms, of the form
h1|(pxa )2 |2i,
h2|(pxa )2 |1i,
h1|(pya )2 |2i,
h2|(pya )2 |1i
(6)
We recall that the momentum operator p = −ih̄∇ is related to the infinitesimal
translation operator T , namely we have, for a generic wavefunction φ,
Tε φ(x) = φ(x + ε) ' φ(x) + εi
∂φ(x)
∂xi
(7)
Suppose to choose a reference system in which the zero-modes |1i and |2i
lie along the x-axis, a distance r apart (see Fig. 1).
The wavefunctions of |1i and |2i are localized and identical. Therefore, if
φ(x) = |1i, then φ(x + rî) = |2i (î: unit vector of the x-axis). For small r we
can write
i
r px φ(x) = φ(x + rî) − φ(x) = |2i − |1i
(8)
h̄
23
Graviton decay
Applying this to the matrix elements in (6) we obtain, for instance
ih̄
ih̄
h̄2
ih̄
h2|px |2i − (h2| − h1|)|2i = − 2
h1|(p ) |2i = − (h2| − h1|)px |2i = −
r
r
r
r
(9)
In the first step the operator px has acted on the left, and in the second step
we have supposed h2|px |2i = 0 (the zero-mode is at rest; its motion can be
taken into account through a Lorentz boost, see Sect. 4).
In conclusion, we find
"
#
x 2
|hΨf in |Hint |Ψin i|2 =
3
h̄5 G
λF
4m4 r4 c4
(10)
Phase space and UV cut-off
After computing the transition matrix elements, in order to obtain the decay
probability per unit time we use the Fermi golden rule. This needs however
to be adapted to the peculiar process we are handling. Since each graviton
can excite several possible zero-modes, and an excited zero-mode passes to one
single final state, we must sum the matrix element not over the final states,
but over the initial states.
The possible initial states of the zero-modes in a time interval dt are contained in a volume V = Acdt, where A is the area on which, on average, the
wave carries a graviton of energy E. This area is related to the wave intensity
by A = (F dt)−1 ; it follows
c
(11)
V = Acdt =
F
The number of possible initial states in this volume is, apart from an adimensional geometrical factor, of the order of V /r3 , where r is the size of the
zero-mode. We need to express r as a function of the energy E of the incoming
graviton. This will be resonant with the excited state of the zero-mode only if
its binding energy, Ezm = Gm2 /r, is equal to E. For a fixed virtual mass m
we find the density of states
dN
d
=
dE
dE
V
r3
=−
3V dr
3c Gm2
=
r4 dE
F r4 E 2
(12)
Finally the decay probability of the graviton per unit time by excitation of a
zero-mode with mass m is found to be, apart from an adimensional geometrical
factor,
"
#
dP
2π
dN
h̄5 E 5
=
|hΨf in |Hint |Ψin i|2
∝ 2 6 16
(13)
dt m
h̄
dE
cG m
It only depends on the fundamental constants h̄, c, G, on the graviton energy
E and on the virtual mass m.
24
Giovanni Modanese
Figure 2: Solution of the cut-off equations (16), (17).
In order to obtain the total probability, the next step is to integrate on
the virtual mass m, with some suitable cut-off. The value of m varies along
with the value of the size r of the zero-mode, keeping the binding energy E
constant and equal to the graviton energy. When m decreases, r decreases too,
therefore the cut-off is of the UV type, in spite of the fact that a small m limit
is usually considered a IR divergence in quantum field theory. (Also note that
in the opposite limit of large r and large m the oscillations in the wave factor
exp(ik α xα ) lead to a suppression of the decay probability.)
Physically, the cut-off is due to the quantum uncertainty on the position of
the two interacting virtual masses. A necessary condition implicit in our calculation is that during one oscillation of the gravitational wave the displacement
of each of the two virtual masses is not larger than their distance, or
v∆t < r ⇒ v < f r
(14)
(Here f is the frequency of the wave and ∆t ∼ 1/f .) From the spacemomentum uncertainty relation, applied to each virtual mass, we obtain
∆p∆x ≥ h̄ ⇒ mvr ≥ h̄
(15)
Therefore it is necessary that
E
m r2 ≥ h̄
h̄
The relation between E and m also holds:
s
m=
E
r
G
(16)
(17)
25
Graviton decay
In the m-r plane, the region allowed by (16) lies over the curve 1 (see Fig.
2), of equation m = h̄2 /Er2 , while at the same time (17) holds (curve 2),
therefore the cut-off is
!1
Eh̄2 5
(18)
mmin =
G2
Replacing this into (13) we finally obtain the decay probability
1
dP
1 = A 2 h̄2 G2 ω 9 5
dt
c
(19)
where A is an adimensional geometric constant of order 1.
4
Conclusions. Outlook
We have found a consistent and finite expression (eq. (19)) for the decay probability of a graviton by interaction with nonlinear gravitational vacuum fluctuations (negative virtual masses). For a numerical estimate we can consider, for
instance, a frequency f ' 10 Hz. (The gravitational waves whose observation
is reported in [1] have frequency in the range 101 – 102 Hz.) We obtain in
' 10−31 s−1 , or a lifetime τ ' 1031 s. This very long lifetime is
this way dP
dt
compatible with the fact that the observed amplitude of the wave in [1] agrees
with the energy released from its supposed source, i.e., there is essentially no
dissipation in the propagation.
An important technical point which we shall address in detail in future
publications is the integration of the probability given by (19) over all possible
Lorentz boosts of the zero-modes. This is a peculiar feature of this decay process and originates from the fact that the decay does not occur by a scattering
with real particles, but with Lorentz-invariant vacuum fluctuations. Preliminary computations show that the integration over Lorentz boosts gives a finite
result and does not affect the dependence of the decay probability on ω and
G; therefore the magnitude order of the lifetime is confirmed to be as given
above.
References
[1] B.P. Abbott et al., Observation of Gravitational Waves from a Binary
Black Hole Merger, Phys. Rev. Lett., 116 (2016), 061102.
https://doi.org/10.1103/physrevlett.116.061102
[2] C.P. Burgess, Quantum Gravity in Everyday Life: General Relativity as
an Effective Field Theory, Living Rev. Rel., 7 (2004).
https://doi.org/10.12942/lrr-2004-5
26
Giovanni Modanese
[3] O. Lauscher and M. Reuter, Is quantum Einstein gravity nonperturbatively renormalizable?, Class. Quant. Grav., 19 (2002), 483-492.
https://doi.org/10.1088/0264-9381/19/3/304
[4] C.S. Unnikrishnan and G.T. Gillies, Quantum gravito-optics: a light
route from semiclassical gravity to quantum gravity, Class. Quant. Grav.,
32 (2015), 145012. https://doi.org/10.1088/0264-9381/32/14/145012
[5] M.J. Clark, Radiation and Gravitation, Chapter In: Advances in the
Interplay between Quantum and Gravity Physics, P.G. Bergmann, V. De
Sabbata Ed.s, Kluwer, 2001, 77-84.
https://doi.org/10.1007/978-94-010-0347-6 4
[6] G. Modanese, General estimate for the graviton lifetime, Phys. Lett. B,
348 (1995), 51-54. https://doi.org/10.1016/0370-2693(95)00149-f
[7] G. Fiore and G. Modanese, General properties of the decay amplitudes for massless particles, Nucl. Phys. B, 477 (1996), 623-651.
https://doi.org/10.1016/0550-3213(96)00346-x
[8] K.A. Olive, et al. (Particle Data Group), Review of Particle Physics,
Chin. Phys. C, 38 (2014), 090001.
https://doi.org/10.1088/1674-1137/38/9/090001
[9] G. Modanese, The vacuum state of quantum gravity contains large virtual masses, Class. Quant. Grav., 24 (2007), 1899-1909.
https://doi.org/10.1088/0264-9381/24/8/001
[10] G. Modanese, Anomalous gravitational vacuum fluctuations which act
as virtual oscillating dipoles, Chapter in Quantum Gravity, R. Sobreiro
Ed., InTech, 2012. https://doi.org/10.5772/35910
[11] M.L. Bertotti and G. Modanese, Microscopic models for welfare measures
addressing a reduction of economic inequality, Complexity, 21 (2015),
89-98. https://doi.org/10.1002/cplx.21669
[12] S. Weinberg, Gravitation and Cosmology: Principles and Applications of
the General Theory of Relativity, Wiley, 1972.
[13] M. Maggiore, Gravitational Waves, Oxford Univ. Press, 2008.
https://doi.org/10.1093/acprof:oso/9780198570745.001.0001
Received: September 25, 2016; Published: November 30, 2016