GAUGE FIELDS AND STRINGS
Notice, again, that now t is not a time but the length of the elastic string.
According to the Boltzmann principle, the classical partition function of
the string is proportional to:
( 1.11)
(p being the inverse temperature), (we have omitted the contribution
from the kinetic energy, since in classical statistical mechanics it factors
out and does not depend on x and x'). Comparison of (1.11) and (1.9)
shows the first analogy between classical statistical mechanics and
quantum mechanics: The transition amplitude for a quantum particle for
the time ( —iT) is equal to the classical partition function for a string of
length T computed at the value of f = l/h.
The second analogy follows from the fact that the quantum partition
function for the particle is given by
= Tv e~^^ and hence:
dx F(x, X, — ißh)
( 1.12)
Therefore our second rule is that in the quantum case the inverse
temperature acts as imaginary time.
Our derivation of these analogies was technical. I feel that there are
deep reasons for them, connected with the properties of space-time.
Although no real explanation exists, I shall give some comments on this
below, when discussing gravity. At the moment our aims are more
modest—we are going to exploit these analogies in concrete problems.
It is quite clear that, although we have derived everything for one
particle, both of our analogies are true for an arbitrary number of
degrees of freedom.
1.2 . Global and Local Symmetries.
Preliminary Description
Elementary particles existing in nature resemble very much excitations
of some complicated medium (ether). We do not know the detailed
structure of the ether but we have learned a lot about effective
lagrangians for its low energy excitations. It is as if we knew nothing
about the molecular structure of some liquid but did know the
Navier-Stokes equation and could thus predict many exciting things.
Clearly, there are lots of different possibilities at the molecular level
Notice, again, that now t is not a time but the length of the elastic string.
According to the Boltzmann principle, the classical partition function of
the string is proportional to:
( 1.11)
(p being the inverse temperature), (we have omitted the contribution
from the kinetic energy, since in classical statistical mechanics it factors
out and does not depend on x and x'). Comparison of (1.11) and (1.9)
shows the first analogy between classical statistical mechanics and
quantum mechanics: The transition amplitude for a quantum particle for
the time ( —iT) is equal to the classical partition function for a string of
length T computed at the value of f = l/h.
The second analogy follows from the fact that the quantum partition
function for the particle is given by
= Tv e~^^ and hence:
dx F(x, X, — ißh)
( 1.12)
Therefore our second rule is that in the quantum case the inverse
temperature acts as imaginary time.
Our derivation of these analogies was technical. I feel that there are
deep reasons for them, connected with the properties of space-time.
Although no real explanation exists, I shall give some comments on this
below, when discussing gravity. At the moment our aims are more
modest—we are going to exploit these analogies in concrete problems.
It is quite clear that, although we have derived everything for one
particle, both of our analogies are true for an arbitrary number of
degrees of freedom.
1.2 . Global and Local Symmetries.
Preliminary Description
Elementary particles existing in nature resemble very much excitations
of some complicated medium (ether). We do not know the detailed
structure of the ether but we have learned a lot about effective
lagrangians for its low energy excitations. It is as if we knew nothing
about the molecular structure of some liquid but did know the
Navier-Stokes equation and could thus predict many exciting things.
Clearly, there are lots of different possibilities at the molecular level