Adaptive Perturbation Theory: QM and Field Theory

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Adaptive Perturbation TheoryQM and Field Theory

• Marvin Weinstein

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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
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The Simple Harmonic Oscillator

• Consider usual harmonic oscillator
• Usual results

variational
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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
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Variational Trickery

• Once again
• Then the Hamiltonian becomes

variational
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Numerical Results

• Trial States
• N0
• N gt 0

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Double Well

• Now consider
• Simple variational fnctn wont work well.
• But can try a shifted Gaussian

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Shifted Formulas

• The shift amounts to rewriting
• The Hamiltonian becomes

Now vary wrt g and c
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Generic Behavior

• Tricritical behavior i.e.,
• 3 minima
• for large mass this
• means 1st order
• phase transition

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The Full Energy Surface
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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 N0 and N1 states.

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Why Solve 2x2 Hamiltonian ?

• Solve the following minimum problem
• Differentiating we get

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l 1 f 2.24

• Now we have two well separated
  minima and no minimum at the origin

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l 1 f 2.24.. Inspect the Saddle

• A close inspection for this case shows
  that now c0 is a local maximum.

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l 1 f 1.73.. Another View

• For smaller f the same is true
  Two separated minima, no minimum
  at 0.

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l 1 f 2 0 Still Two Minima ?

• At first this seems surprising, but
  on reflection it is correct.
• .

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Tunneling Computation (even)

• Trial state is now
• The change in sign is because c
  -gt - c

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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

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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

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On To Field Theory

• Hamiltonian in momentum space is
• Introduce variational parameters

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Expectation Value of Hamiltonian

• Taking the expectation value of H in the
  vacuum state
• Differentiating wrt
• So

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Solving

• This clearly has a solution
• Actually this is an equation for m

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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

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What Is The Change In Energy ?

• To get a formula for what this does consider
• This is solved by iteration

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Taking Expectation Value

• Lowest energy is pole in z so
• So we look for a zero of the denominator

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Solving for z

• Once again we need to solve
• Lets redefine z
• So the equation becomes
• in the limit

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This Equation Can Be Solved Iteratively

• Define a sequence

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How Well Does This Work ?

• For a 2x2 Matrix
• Let and and
  let vary

1 Iteration
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More Iterations

• Three Iterations

Error
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Twelve Iterations
The same is true formomentum integralswith
about the samerate of convergence.Thus, the
answer canalways be expressedas a continued
fraction.
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Back To Field Theory

• With these observations with the 4-particle
  contribution the vacuum energy is of the form

Diverges like L4
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Minimizing

• Differentiating wrt yields eqn of form
• or

Divergeslike L2
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Wave Function Renormalization

• Usual prescription
• We can by convention put this in a form
• by rescaling

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What About Coupling Constant Ren ?

• Coupling constant renormalization isnt
  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.

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As Before Use Resolvent Operator

• The k0 two particle energy

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Bug or Feature ?

• One particle state isnt 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 ?

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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