QUARK CONFINEMENT
83
walls) or whether it will stay at rest (as in the case of a superfluid). Let
us go to the coordinate system where the walls are at rest. This is
achieved by a Galilean transformation
T(x)*T(jc)
(5.47)
after which
<#>■
= <^o> “
+ Nmv^/2
(5.48)
Here
Po =
. 1 ^
V {/+ _ ipdjc
1 dx
is the total momentum. We see that if before the transformation the
liquid was at rest (Pq = 0, j^q = Eq) we have to expect that the change
(5.47) will add to the Hamiltonian a i;^-term. If the liquid moves then
J^Q = Eq Nmv^/2; Pq = -^Nmv and the substitution (5.47) should
add nothing to < Jfo>- If recall now that the only infrared-important
quantity in "^He is (p(x\ the phase of 4^(jc), we conclude that the
existence of superfluidity manifests itself in the response of our system
to the transformation:
(p(x) (p — mvx
(5.49)
Normal liquid is indifferent to such a transformation, while the
superfluid does react. We have also seen that such a response is possible
exclusively because of the pole terms in (5.44). The dielectric constant in
a gauge system was shown to be a precise analogue of the superfluid
density Ps- This analogy can be extended further in order to include
crystals which, as we saw, are described by the same kind of actions
depending on uj^x). In this case we have to examine the response of the
system to a symmetric traceless field defined by:
(5.50)
= 0). Again, nonzero response to
(which is a kind of external
gravitational field, just as could have been considered as an external
electromagnetic field), is possible in the longranged phase, and is
described by the shear modulus
(5.51)
83
walls) or whether it will stay at rest (as in the case of a superfluid). Let
us go to the coordinate system where the walls are at rest. This is
achieved by a Galilean transformation
T(x)*T(jc)
(5.47)
after which
<#>■
= <^o> “
+ Nmv^/2
(5.48)
Here
Po =
. 1 ^
V {/+ _ ipdjc
1 dx
is the total momentum. We see that if before the transformation the
liquid was at rest (Pq = 0, j^q = Eq) we have to expect that the change
(5.47) will add to the Hamiltonian a i;^-term. If the liquid moves then
J^Q = Eq Nmv^/2; Pq = -^Nmv and the substitution (5.47) should
add nothing to < Jfo>- If recall now that the only infrared-important
quantity in "^He is (p(x\ the phase of 4^(jc), we conclude that the
existence of superfluidity manifests itself in the response of our system
to the transformation:
(p(x) (p — mvx
(5.49)
Normal liquid is indifferent to such a transformation, while the
superfluid does react. We have also seen that such a response is possible
exclusively because of the pole terms in (5.44). The dielectric constant in
a gauge system was shown to be a precise analogue of the superfluid
density Ps- This analogy can be extended further in order to include
crystals which, as we saw, are described by the same kind of actions
depending on uj^x). In this case we have to examine the response of the
system to a symmetric traceless field defined by:
(5.50)
= 0). Again, nonzero response to
(which is a kind of external
gravitational field, just as could have been considered as an external
electromagnetic field), is possible in the longranged phase, and is
described by the shear modulus
(5.51)