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Adaptive Perturbation Theory:
QM and Field Theory
Marvin Weinstein
Two Topics
Quantum mechanics
What your grandmother
never told you about
perturbation theory. But
should have !
Field Theory
Apply the QM tricks to the case of scalar
field theory
The Simple Harmonic Oscillator
Consider usual harmonic oscillator
Usual results
variational
The Anharmonic Oscillator
The Hamiltonian is
The usual problems
Perturbation expansion diverges due to
N! growth of the terms
New result: Adaptive perturbation theory
converges for all couplings and N.
Variational Trickery
Once again:
Then the Hamiltonian becomes
variational
Numerical Results
Trial States
N=0 :
N > 0:
Double Well
Now consider
Simple variational fnctn
won’t work well.
But can try a shifted Gaussian
Shifted Formulas
The shift amounts to rewriting
The Hamiltonian becomes
Now vary wrt g and c
Generic Behavior
Tricritical behavior; i.e.,
3 minima
for large mass this
means 1st order
phase transition
The Full Energy Surface
Doing Better
Exploit the linear term in creation and
annihilation operators.
In other words use a trial state of the form
Varying is the same as diagonalizing the 2x2-matrix
obtained by restricting the Hamiltonian to the N=0 and
N=1 states.
l = 1 f = 2.24…
Now we have two
well separated
minima and no
minimum at
the origin
l = 1 f = 2.24.. Inspect the Saddle
A close inspection for
this case shows that
now c=0 is a local
maximum.
l = 1 f = 1.73.. Another View
For smaller f
the same is true
Two separated
minima, no minimum
at 0.
l = 1 f = 0 Still Two Minima ?
2
At first this seems
surprising, but
on reflection it is
correct.
.
Tunneling Computation (even)
Trial state is now
The change in
sign is because
c -> - c
Tunneling Computation (odd)
Now the trial state is
Re-minimize and
once again we are
good for ground-state
and first excited state
to 0.009 %
Large N
Still two minima, but the distance to the
minimum stops growing.
However the width of the wave-function
keeps growing. More importantly these
expection value of the x4 term keeps growing,
which implies the tunneling effect increases.
Sphaleron is approximately when the splitting
is no longer exponentially suppressed
On To Field Theory
Hamiltonian in momentum space is
Introduce variational parameters
Expectation Value of Hamiltonian
Taking the expectation value of H in the
vacuum state:
Differentiating wrt
So
Solving
This clearly has a solution
Actually this is an equation for m
Capturing Wave-Function Renormalization
Choose as the variational state
We need to get the change in the vacuum
energy due to this piece of Hamiltonian:
What Is The Change In Energy ?
To get a formula for what this does consider
This is solved by iteration
Taking Expectation Value
Lowest energy is pole in z so
So we look for a zero of the denominator
Solving for z
Once again we need to solve
Lets redefine z
So the equation becomes
in the limit
This Equation Can Be Solved Iteratively
Define a sequence
How Well Does This Work ?
For a 2x2 Matrix
Let
and let
and
vary
1 Iteration
More Iterations
Three Iterations
% Error
Twelve Iterations
The same is true for
momentum integrals
with about the same
rate of convergence.
Thus, the answer can
always be expressed
as a continued
fraction.
Back To Field Theory
With these observations with the 4-particle
contribution the vacuum energy is of the form:
Diverges like L4
Minimizing
Differentiating wrt
yields eqn of form
or
Diverges
like L2
Wave Function Renormalization
Usual prescription
We can by convention put this in a form
by rescaling
What About Coupling Constant Ren ?
Coupling constant renormalization isn’t
required, just a choice of coupling constant.
Question: What do we hold fixed ?
My choice is the energy of the zero
momentum two-particle state.
This immediately shows why this theory is
trivial in four dimensions.
As Before Use Resolvent Operator
The k=0 two particle energy
Bug or Feature ?
One particle state isn’t boost invariant
Parton picture ?
Have to both redo the one-particle variation and
add extra particles to correct the wrong
k – dependence
Non-covariant effects ?
New counter-terms ?
New Lattice Approximation ?
IDEA: Now that the parameters are determined
we can, in the presence of a cut-off inverse
Fourier transform back to a lattice theory
with coefficients which depend on the
parameters. THEN DO CORE