THE INTERNATIONAL SERIES OF MONOGRAPHS ON PHYSICS ...

6
ADVANTAGES AND DRAWBACKS OF EFFECTIVE THEORY
6.1 Non-locality in effective theory
6.1.1 Conservation and covariant conservation
As is known from general relativity, the equation T µ ν;µ =0or
µ √
∂µ T ν −g =
√ −g
2 T αβ ∂νgαβ (6.1)
does not represent any conservation in a strict sense, since the covariant derivative
is not a total derivative (Landau and Lifshitz 1975). In superfluid 4He it
acquires the form
µ √ d
∂µ T ν −g =
3p (2π¯h) 3 f∂ν ˜ E = P M i ∂νv i
d
s +
3p (2π¯h) 3 f|p|∂νc. (6.2)
This does not mean that energy and momentum are not conserved in superfluids.
One can check that the momentum and the energy of the whole system
(superfluid vacuum + quasiparticles)
d 3
d
r mnvs +
3
p
pf , d
(2π¯h) 3 3
m
r
2 nv2
d
s + ɛ(n)+
3p (2π¯h) 3 ˜
Ef ,
(6.3)
are conserved. For example, for the density of the total momentum of the liquid
one has the conservation law
with the following stress tensor:
Πik = Pivsk +vsiP M
∂ɛ
k +δik n
∂n +
∂t(Pi)+∇kΠik =0,Pi = mnvsi + P M i , (6.4)
d3
p ∂E
f − ɛ +
(2π¯h) 3 ∂n
d3p ∂E
pkf .
(2π¯h) 3 ∂pi
(6.5)
Equation (6.4) together with the corresponding equation for the density of the
total energy can be written in the form
µ
T ν(vacuum) + √ −gT µ ν(matter) =0. (6.6)
∂µ
This is the true conservation law for the energy and momentum, while the
covariant conservation law (6.1) or (6.2) simply demostrates that the energy and
momentum are not conserved for the quasiparticle subsystem alone: there is an