The “a” theorem
and the Markov property of the vacuum
Eduardo Testé
Instituto Balseiro, Centro Atómico Bariloche, S.C. de Bariloche, Argentina
with Horacio Casini and Gonzalo Torroba
based on
the entropic “c” and “F” theorems (Casini, Huerta)
1.
SCFT (r) = µd
entanglement
entropy of a sphere
d
r
2
2
+ ...
{
( 1)d/2
( 1)(d
1
4 a log(r/✏)
1)/2
F
ordered under RG flows?
the entropic “c” and “F” theorems (Casini, Huerta)
1.
SCFT (r) = µd
entanglement
entropy of a sphere
2.
d
r
2
2
+ ...
{
( 1)d/2
( 1)(d
1
4 a log(r/✏)
1)/2
F
ordered under RG flows?
The Strong Subadditivity (SSA) inequality
S(A) + S(B) > S(A ^ B) + S(A _ B)
For the vacuum, which subalgebras saturates the SSA?
S(A) + S(B) = S(A ^ B) + S(A _ B) ?
For the vacuum, which subalgebras saturates the SSA?
S(A) + S(B) = S(A ^ B) + S(A _ B) ?
(Markov state)
null plane
A
A^B
B
in general
For the vacuum, which subalgebras saturates the SSA?
S(A) + S(B) = S(A ^ B) + S(A _ B) ?
(Markov state)
null cone
null plane
A
A^B
A^B
B
A
in general
B
for a CFT
Application: entropic proof of the “a” theorem
S(A) + S(B) > S(A ^ B) + S(A _ B)
S(A) + S(B) = S(A ^ B) + S(A _ B)
vacuum of the
RG running
QFT
vacuum of the
UV CFT
(Markovian)
A^B
A
B
Application: entropic proof of the “a” theorem
S(A) + S(B) > S(A ^ B) + S(A _ B)
S(A) + S(B) = S(A ^ B) + S(A _ B)
S(A) +
S(B) >
S(A ^ B) +
vacuum of the
RG running
QFT
vacuum of the
UV CFT
(Markovian)
S(A _ B)
A^B
A
B
Application: entropic proof of the “a” theorem
S(A) +
take symmetric form of this
S(B) >
S(A ^ B) +
S(A _ B)
as in the F theorem
The differences in local
curvatures are UV and cancels
SUV CFT
=
SRG QFT
S̃ !
No angle
contribution
problem
00
r S (r)
(d
S=
SUV CFT
SRG QFT
3) S (r) 6 0
0
Application: entropic proof of the “a” theorem
00
r S (r)
(d
3) S (r) 6 0
0
SCFT (r) = µ2 r2
4 a log(r/✏)
d=4
aU V > aIR
Application: entropic proof of the “a” theorem
00
r S (r)
d=2
cU V > cIR
(d
3) S (r) 6 0
0
SCFT (r) = µ2 r2
d=3
FU V > FIR
4 a log(r/✏)
d=4
aU V > aIR
Application: entropic proof of the “a” theorem
00
r S (r)
d=2
cU V > cIR
(d
3) S (r) 6 0
0
SCFT (r) = µ2 r2
d=3
FU V > FIR
4 a log(r/✏)
d=4
aU V > aIR
unified picture of RG irreversibility
Application: entropic proof of the “a” theorem
00
r S (r)
d=2
cU V > cIR
(d
3) S (r) 6 0
0
SCFT (r) = µ2 r2
d=3
FU V > FIR
4 a log(r/✏)
d=4
aU V > aIR
unified picture of RG irreversibility
Thank you
(I will be outside with a poster)