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Physics Letters B 772 (2017) 699–702
Contents lists available at ScienceDirect
Physics Letters B
www.elsevier.com/locate/physletb
Scattering and momentum space entanglement
Gianluca Grignani
a
, Gordon W. Semenoff
b,∗
a
Department of Physics and Geology, University of Perugia, I.N.F.N. Sezione di Perugia, Via Pascoli, I-06123 Perugia, Italy
b
Department of Physics and Astronomy, University of British Columbia, 6224 Agricultural Road, Vancouver, British Columbia, V6T 1Z1, Canada
a r t i c l e i n f o a b s t r a c t
Article history:
Received
11 July 2017
Accepted
14 July 2017
Available
online 20 July 2017
Editor:
M. Cveti
ˇ
c
We derive a formula for the entanglement entropy of two regions in momentum space that is generated
by the scattering of weakly interacting scalar particles. We discuss an example where weak interactions
entangle momentum scales above and below an infrared cutoff.
© 2017 The Author(s). Published by Elsevier B.V. This is an open access article under the CC BY license
(http://creativecommons.org/licenses/by/4.0/). Funded by SCOAP
3
.
Recently, some insight into the structure of interacting quantum
field theories, and in particular conformal field theories [1] has
been gathered by studying quantum information theoretic issues
such as the entanglement entropy of degrees of freedom residing
in different regions of coordinate space. One interesting offshoot
of this set of ideas is the notion of entanglement entropy in mo-
mentum
space. From one point of view, there would seem to be
some natural questions to address. The separation of the degrees
of freedom in a quantum field theory according to the distance
scale is a well known ingredient of the Wilson approach to the
renormalization group where one might wonder whether, even af-
ter
the construction of effective field theory describing physics at
a given distance scale, subtle quantum effects could still correlate
physics at that scale with physics at other far disparate scales [2].
On the other hand, interacting quantum field theory is not local
in momentum space and the precise meaning of the degrees of
freedom residing in a region of momentum space is a subtle ques-
tion.
For example, there exist computations of the entanglement
of momentum space regions in weakly coupled field theory using
perturbation theory and the interaction picture [2,3]. Those com-
pute
the entanglement of interaction picture degrees of freedom.
The interaction picture fields evolve in time as free fields, but the
quantum field theory vacuum is a nontrivial correlated state of
those free fields and the correlations include quantum entangle-
ment.
However, the physical interpretation of interaction picture
fields is obscure. They are not the degrees of freedom which cou-
ple
directly to probes or compute correlation functions or S-matrix
elements, the more relevant degrees of freedom being the Heisen-
berg
fields. However, Heisenberg fields obey field equations which
are not local in momentum space, in fact, the interactions have
*
Corresponding author.
E-mail
address: gordonws@phas.ubc.ca (G.W. Semenoff).
infinite range there and it is again not clear that there is a mean-
ingful
way to separate them into operators which live in different
regions of momentum space. An attractive alternative in this re-
gard
are the Lehman–Symanzik–Zimmerman (LSZ) in- or out-fields
which are large negative or positive time limits of the Heisenberg
fields. The in- and out-fields are free fields and, like the interaction
picture fields, their momentum states are not coupled by their field
equations. There is thus a well-defined meaning of in-fields resid-
ing
in particular regions of momentum space. On the other hand,
their vacuum coincides with the interacting quantum field theory
vacuum, and it is not a correlated state of different in-field mo-
menta.
To find such correlations, we must find a way to input the
dynamics of the quantum field theory.
For
in- and out-fields, the interactions are encapsulated in the
S-matrix of the quantum field theory. An in-state evolves into an
out-state which is a superposition of in-states. The individual co-
efficients
in that superposition are the transition amplitudes, or
S-matrix elements. Thus, generally, the out-states are entangled
states of in-fields and one might naturally ask the question of
whether we can quantify the entanglement of in-field degrees of
freedom which are contained in an out-state, including in-fields
that occupy different regions of momentum space. We shall find
that the answer to this question elicits formulae with ingredients
very similar to, but with details differing from those derived in the
interaction picture [2,3].
In
this paper, we will develop some formulae for computing
momentum space entanglement of the in-fields which are con-
tained
in a out-state in the simple example of a weakly self-
interacting
scalar φ-four theory in four dimensions. This issue has
already been explored in the context of entanglement that is gen-
erated
by scattering events and formulae very similar to what we
derive are already known [4–7] in the context of scattering the-
ory.
Here, we will give an alternative derivation of these formulae.
Our approach yields a general formula (13) for the reduced den-
http://dx.doi.org/10.1016/j.physletb.2017.07.030
0370-2693/
© 2017 The Author(s). Published by Elsevier B.V. This is an open access article under the CC BY license (http://creativecommons.org/licenses/by/4.0/). Funded by
SCOAP
3
.
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