# Second Quantization of Conserved Particles 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 ::)?

p42c4 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 Eq. 4.22 For Fermions

It also gives the right particle exchange statistics. Consider Fermions in the 1,3,4 and 6th 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

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