Tensor product and direct sum  Schmotons  John Baez: A lower

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: e^{ikx-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:

• Coh1(c+ib) = e^ibq e^-icp |0>
• Coh1(c+ib) = iota e^-(c2+b2)/2 SUM_n ((c+ib)ⁿ/ sqrt(n!)) |n>
• Coh1(c+ib) = K exp(ibx - (x-c)²)

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

Coherent states, final version:

• Coh(c+ib) = e^-icp+ibq |0>
• Coh(c+ib) = e^-(c2+b2)/2 SUM_n ((c+ib)ⁿ/ sqrt(n!)) |n>
• Coh(c+ib) = K exp(ibx - (x-c)²)

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:

e^A+B = e^Ae^B e^-[A,B]/2

3/10/1998