Second Quantization of Conserved Particles Electrons, 3He, 4He, etc. And of Non-Conserved Particles Phonons, Magnons, Rotons…

We Found for Non-Conserved Bosons E.g., Phonons that we can describe the system in terms of canonical coordinates We can then quantize the system And immediately second quantize via a canonical (preserve algebra) transform

We create our states out of the vacuum And describe experiments with Green functions With

Creation of (NC) Particles at x We could Fourier transform our creation and annihilation operators to describe quantized excitations in space poetic license This allows us to dispense with single particle (and constructed MP) wave functions

We saw, the density goes from And states are still created from vacuum These operators can create an N-particle state With conjugate Most significantly, they do what we want to! Think

That is, they take care of the identical particle statistics for us I.e., the operators must And the Slater determinant or permanent is automatically encoded in our algebra

Second Quantization of Conserved Particles For conserved particles, the introduction of single particle creation and annihilation operators is, if anything, natural In first quantization,

Then to second quantize The density takes the usual form, so an external potential (i.e. scalar potential in E&M) And the kinetic energy

The full interacting Hamiltonian is then It looks familiar, apart from the two ::, they ensure normal ordering so that the interaction acting on the vacuum gives you zero, as it must. There are no particle to interact in the vacuum Can I do this (i.e. the ::)?

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The Algebra Where + is for Fermions and – for Bosons Here 1 and 2 stand for the full set of labels of a particle (location, spin, …)

Transform between different bases Suppose we have the r and s bases Where I can write (typo) If this is how the 1ps transform then we use if for operators x or k (n)

With algebra transforming as I.e. the transform is canonical. We can transform between the position and discrete basis Where is the nth wavefunction. If the corresponding destruction operator is just

Is this algebra right? It does keep count Since – F [ab,c]=abc-cab + acb-acb =a{b,c}-{a,c}b – B [ab,c]=abc-cab + acb-acb =a[b,c]+[a,c]b – For Fermions Eq. 4.22

It also gives the right particle exchange statistics. Consider Fermions in the 1,3,4 and 6 th one particle states, and then exchange 4 6 Perfect!

And the Boson state is appropriately symmetric 3 hand written examples (second L4 file)

Second Quantized Particle Interactions The two-particle interaction must be normal ordered so that Also hw example