Second Quantization of Conserved Particles

Electrons, 3He, 4He, etc.

And of Non-Conserved ParticlesPhonons, 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 <x|p>

• 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

– • For Fermions

Eq. 4.22

• 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

• Also hw example