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Appendix: Notational Conventions

Two conventions for this HTML version:  sq2 = sqrt(2), and we use w for omega, the frequency or energy.

Plane wave with momentum k and energy w: eikx-iw t.

Metric d tau2 = dt2 - dx2 - dy2 - dz2
Momentum p = -i d/dx
Position q = ``multiply by x''
Hamiltonian H = i d/dt
Annihilator a = (q+ip)/sq2
Creator a* = (q-ip)/sq2
  q = (a+a*)/sq2 
  p = (a-a*)/(i sq2)
commutator [A,B] = AB-BA
  dA/dt = i[H,A]
  dA/dx = -i[p,A]
product rule [s,AB] = [s,A]B + A[s,B]
a*a is also called the number operator, sometimes denoted N.

Coherent states, take 1:

where |iota|=1, the last equation gives Coh1 as a complex wavefunction, and K is a normalization factor.

Coherent states, final version:

If z = c+ib, then e-iHt Coh(z) = e-it/2 Coh(ze-it). Here e-it/2 represents the ``vacuum energy''.

For the full-blown version of the Baker-Campbell-Hausdorff formula, see the postscript version of these note.

Baker-Campbell-Hausdorff formula, special case: if [A,B] commutes with both A and B, then:

eA+B = eAeB e-[A,B]/2


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Next:  Tensor product and direct sum Up: Schmotons Previous: John Baez: A lower 


Michael Weiss

3/10/1998