Let and be time-independent in the Schrödinger picture.
Then, in the Heisenberg picture,
consider the following expectation value
with respect to thermodynamic equilibium
(as found in Green’s functions for example):
Where the “simple” Hamiltonian is time-independent.
Suppose a (maybe time-dependent) “difficult” is added,
so that the total Hamiltonian is .
Then it is easier to consider the expectation value
in the interaction picture:
Where is the time evolution operator of .
In front, we have ,
while is an exponential of an integral of , so we are stuck.
Keep in mind that exponentials of operators
cannot just be factorized, i.e. in general
To get around this, a useful mathematical trick is
to use an imaginary time variable instead of the real time .
Fixing a , we “redefine” the interaction picture along the imaginary axis:
Ironically, is real; the point is that this formula
comes from the real-time definition by replacing .
The Heisenberg and Schrödinger pictures can be redefined in the same way.
In fact, by substituting ,
all the key results of the interaction picture can be updated,
for example the Schrödinger equation for becomes:
And the interaction picture’s time evolution operator
turns out to be given by:
Where is the
with respect to .
This operator works as expected:
Where is related to
the Schrödinger and Heisenberg pictures as follows:
It is interesting to combine this definition
with the action of time evolution :
Rearranging this leads to the following useful
alternative expression for :
Returning to our initial example,
we can set and ,
Using the easily-shown fact that
we can therefore rewrite the thermodynamic expectation value like so:
We now introduce a time-ordering ,
letting us reorder the (bosonic) -operators inside,
and thereby reduce the expression considerably:
If we now define as the expectation value with respect
to the unperturbed equilibrium involving only ,
we arrive at the following way of writing this time-ordered expectation:
For another application of imaginary time,
see e.g. the Matsubara Green’s function.
- H. Bruus, K. Flensberg,
Many-body quantum theory in condensed matter physics,