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Analogy between a classical discrete system and non-classical continuous system

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Most introduction textbooks about quantum fieldtheory start with a discrete classical harmonic oscillator and then looks at it in the continuous quantized case (quantized field). This leads to the first quantum fieldtheory for a spin 0 field. Now from there it usually goes on to the spin 1/2, spin 1 field and so on, but leaves out the discrete classical versions of these fields.

What is the analog in the discrete classical form? It should be something like:

To get spin 0: discrete and classical harmonic oscillator -> continuous and non-classical harmonic oscillator

To get spin 1/2: discrete and classical vector field (not sure?) -> continuous and non-classical vector field

To get spin 1: discrete and classical tensor field (?) -> continuous and non-classical tensor field

For example the discrete und classical form of the Klein-Gordon field is the harmonic oszillator. What's the analogy with the Dirac field and so on?

asked Jun 18, 2015 in Theoretical Physics by WolfInSheepSkin (-40 points) [ no revision ]

What do you mean by "discrete"? A classical harmonic oscillator isn't discrete in any sense - or do you mean a system of classical oscillators with discrete spectrum?

I mean degrees of freedom. This is always discussed in introductions to QFT.

1 Answer

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The free quantum field theories are classified by representation spaces of the irreducible unitary representations of the Poincare group, which serve as the corresponding 1-particle spaces. Their classical counterparts are the symplectic spaces defined on the coadjoint orbits of the Poincare group. In the massive case, this is for spin 0 the cotangent space $R^6$ of one classical spinless particle in $R^3$, and for spin $>0$ the space $R^6\times S^2$ of a classical spinning particle; classically, the spin can take any positive value, not only half-integral ones.

The quantum harmonic oscillator is the special case of a single mode of a bosonic free quantum field theory (typically a mode with fixed momentum, hence energy $E=\hbar\omega$, giving the frequency $\omega$); its $N$th excitation is the corresponding $N$-particle state. The fixed momentum modes themselves are obtained by a spectral analysis of the solutions of the wave equation of the corresponding 1-particle representation, hence of the Klein-Gordon equation for spin 0, the Maxwell equations for spin 1, etc..

For fermions, one doesn't get a harmonic oscillator for each mode but (because of the exclusion principle) a qubit, a quantum 2-state system. For spin 1/2, each fixed momentum modes is obtained by a spectral analysis of the solutions of the Dirac wave equation.

The Ising model is a lattice version of a nonrelativistic spinor field theory; it has at each lattice point a qubit, i.e., the fermionic version of the harmonic oscillator. Spin waves are the lattice analogues of the fixed momentum modes. There are many variants of the Ising model since you can put on each lattice point not just a spin degree of freedom but more complicated objects, and you can also decorate the edges (corresponding to interactions) in more complex ways. Finally, one can replace the lattice by more general graphs or related topological networks.

answered Jun 18, 2015 by Arnold Neumaier (12,355 points) [ revision history ]
edited Jun 19, 2015 by Arnold Neumaier

Hi. Thx for the answer. But I didn't demand a harmonic oscillator. I just want to know if there is a similar analog. I asked at physics exchange, and a user talked about spin waves or fermions on a lattice or a so called Ising model. Any comment on this?

See the update of my answer.

Alright. Thank you. Is there a minimal simple model that could be used in a pedagogical introduction like I thought of?

A minimal spin 1/2 model for pedagogical purposes could be a 1-dimensional Ising model on a polygon. The pentagon has a 32-dimensional state space, and everything of interest can probably be demonstrated explicitly, though spin waves still look a bit rugged - to get these look nice you'd need more vertices.

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