1. Adaptive Perturbation Theory: QM and Field Theory
Marvin Weinstein

2. Two Topics Quantum mechanics Field Theory
What your grandmother never told you about perturbation theory. But should have !
Apply the QM tricks to the case of scalar field theory

3. The Simple Harmonic Oscillator
Consider usual harmonic oscillator
Usual results
variational

4. The Anharmonic Oscillator
The Hamiltonian is
The usual problems
New result: Adaptive perturbation theory converges for all couplings and N.
Perturbation expansion diverges due to N! growth of the terms

5. Variational Trickery Once again: Then the Hamiltonian becomes

6. Numerical Results
Trial States
N=0 :
N > 0:

7. Double Well Now consider Simple variational fnctn won’t work well.
But can try a shifted Gaussian

8. Shifted Formulas The shift amounts to rewriting
The Hamiltonian becomes
Now vary wrt g and c

9. Generic Behavior Tricritical behavior; i.e., 3 minima
for large mass this
means 1st order
phase transition

10. The Full Energy Surface

11. 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.

12. l = 1 f = 2.24…
Now we have two well separated minima and no minimum at the origin

13. l = 1 f = 2.24.. Inspect the Saddle
A close inspection for this case shows that now c=0 is a local maximum.

14. l = 1 f = Another View
For smaller f the same is true Two separated minima, no minimum at 0.

15. l = 1 f 2 = 0 Still Two Minima ?
At first this seems surprising, but on reflection it is correct. .

16. Tunneling Computation (even)
Trial state is now
The change in sign is because c -> - c

17. Tunneling Computation (odd)
Now the trial state is
Re-minimize and once again we are good for ground-state and first excited state to %

18. 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

19. On To Field Theory Hamiltonian in momentum space is
Introduce variational parameters

20. Expectation Value of Hamiltonian
Taking the expectation value of H in the vacuum state:
Differentiating wrt So

21. Solving
This clearly has a solution
Actually this is an equation for m

22. 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:

23. What Is The Change In Energy ?
To get a formula for what this does consider
This is solved by iteration

24. Taking Expectation Value
Lowest energy is pole in z so
So we look for a zero of the denominator

25. Solving for z Once again we need to solve Lets redefine z
So the equation becomes
in the limit

26. This Equation Can Be Solved Iteratively
Define a sequence

27. How Well Does This Work ? For a 2x2 Matrix Let and and let vary
1 Iteration

28. More Iterations
Three Iterations
% Error

29. 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.

30. Back To Field Theory
With these observations with the 4-particle contribution the vacuum energy is of the form:
Diverges like L4

31. Minimizing
Differentiating wrt yields eqn of form or
Diverges like L2

32. Wave Function Renormalization
Usual prescription
We can by convention put this in a form
by rescaling

33. 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.

34. As Before Use Resolvent Operator
The k=0 two particle energy

35. 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 ?

36. 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