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  Quantizing Klein-Gordon via Lie Groups

+ 3 like - 0 dislike

I'm trying to understand second quantization of the Klein-Gordon equation, as explained in, say, standard books like Peskin and Schroder, but using the language of Lie (representation) theory. In a sense I just want to be able to attach the correct mathematical words to each step of the process, e.g. this step is talking about a Heisenberg Lie algebra, but we interpret it as embedded in the Lie algebra of a Poincare group because of another step, that step goes to the dual Lie algebra, etc... but it's a bit confusing as to what's actually going on.

Motivating this question are Woit's notes on QM/QFT, pages 438, 441, 446, I've given the motivating quotes below.

My hope is that the process is something like:

  • Manifold: Minkowski Space
  • Lie Group: Poincare Group
  • Lie Algebra: Poincare Algebra (this gives us a Lie-algebra representation of the (Poincare) Lie Group).
  • Universal Enveloping Algebra: (Just to be able to form a Casimir)
  • Casimir Element: Klein-Gordon Operator
  • Casimir Eigenvalue Equation: Klein-Gordon Equation where eigenfunctions are just elements of the Poincare Lie algebra, and these are irreducible representations of the Poincare group, (which explains why wave functions are operators and avoids a classical scalar field interpretation)
  • Solution Method: Go to the dual Lie algebra (take a Fourier transform), solve as an algebraic equation, then invert to find $\hat{\psi}(x)$, but since this is a representation of the dual Poincare group (page 438 comment below), we need to randomly form quadratic combinations of the fields $\hat{\psi}(x), \hat{\pi}(x)$ and integrate over them (to get the Hamiltonian) and then commute this with those $\hat{\psi}(x), \hat{\pi}(x)$ fields (even though they all represent different spaces) and this gives us a solution to the Klein-Gordon equation... This is supposed to be motivated by thinking about intertwining operators (to find equivalent representations), but it really makes no sense to me...

Note this process completely ignores classical field theory, it is group theoretical only, the classical wave equation only arises as the Casimir of the Poincare group, spitting out irreducible representations, with function expressed in a basis of these gives a wave function, and the structure of the whole process just follows the same process you follow in Lie theory when looking at a Lie group to, say, prove it is simple/semi-simple...

However Woit's notes seem to say something different, pages 438, 441 and 446 seem to indicate the process is more like:

  • Manifold: Phase space of solutions (scalar fields) to classica Klein-Gordon PDE
  • Lie Group: Poincare Group (acting on functions in the phase space)
  • Lie Algebra: Not the Poincare Algebra (acting on functions in the phase space), but instead the representation of a (random) Heisenberg algebra derived from the symplectic structure of the phase space which we (randomly) use to find irreducible representations of the Poincae group, by applying the Casimir of a totally different Lie algebra coming from the Poincare group,
  • Universal Enveloping Algebra: ...
  • Casimir Element: Not the Klein-Gordon Operator acting on functions in the phase space coming from the Poincare group Lie algebra, but instead the Klein-Gordon operator acting on "unitary representation of a Heisenberg Lie algebra on $M \oplus R$, where M is the dual of the space of solutions of the Klein-Gordon equation",
  • Casimir Eigenvalue Equation: Eigenfunctions are representations of the solutions of the classical Klein-Gordon equation, the dual of the space of solutions of the Klein-Gordon equation, but (p.446) they are also irreducible representations of the Poincare group somehow.
  • Solution Method: Treat the wave function $$\hat{\phi}(x) = \dfrac{1}{(2\pi)^{3/2}} \int_{\mathbb{R}^3} (a(\vec{p})e^{i \vec{p} \cdot \vec{x}} + a^{+}(\vec{p})e^{- i \vec{p}\cdot \vec{x}})\dfrac{d^3\vec{p}}{\sqrt{2\omega_{\vec{p}}}} $$ along with the commutation relations $$[\hat{\phi},\hat{\pi}] = ..., [\hat{\phi},\hat{\phi}] = ..., $$ as a unitary representation of a Heisenberg Lie algebra on $M \oplus R$, where M is the dual of the space of solutions of the Klein-Gordon equation. Then to find a representation of the Poincare group from this representation of a completely different space, we form quadratic combinations of $\hat{\phi},\hat{\pi}$ and integrate to form the Hamiltonian (integrating so that we are representing the original Poincare group I guess) and then randomly commuting the Hamiltonian representing the original Poincare group with representations of a dual Lie algebra.

This process assumes the classical Klein-Gordon theory (for no reason), seems to mix up the spaces on which everything is done, and use this crazy Heisenberg Lie algebra coming from a representation of a function space to describe the Poincare group acting on the original phase space. It should look obvious and coherent, I just don't see it.

The quotes motivating all this are:

Page 438:

The real scalar quantum field operators are the operator-valued distributions defined by
$$\hat{\phi}(x) = \dfrac{1}{(2\pi)^{3/2}} \int_{\mathbb{R}^3} (a(\vec{p})e^{i \vec{p} \cdot \vec{x}} + a^{+}(\vec{p})e^{- i \vec{p}\cdot \vec{x}})\dfrac{d^3\vec{p}}{\sqrt{2\omega_{\vec{p}}}} $$
The commutation relations
$$[\hat{\phi},\hat{\pi}] = ..., [\hat{\phi},\hat{\phi}] = ..., $$
can be interpreted as the relations of a unitary representation of a Heisenberg Lie algebra on $M \oplus R$, where M is the dual of the space of solutions of the Klein-Gordon equation.

Page 441:

The complex scalar quantum field operators are the operator-valued distributions that provide a representation of the infinite-dimensional Heisenberg algebra given by the linear functions on the phase space of solutions to the complexified Klein-Gordon equation. This representation will be on a state space describing both particles and antiparticles.

which interpret the wave functions $\hat{\phi}$ as representations of a Heisenberg Lie algebra on the space of solutions of Klein-Gordon, and

Page 446:

Just as for non-relativistic quantum fields, the theory of free relativistic scalar quantum fields starts by taking as phase space an infinite dimensional space of solutions of an equation of motion. Quantization of this phase space proceeds by constructing field operators which provide a representation of the corresponding  Heisenberg Lie algebra, using an infinite dimensional version of the Bargmann- Fock construction. In both cases the equation of motion has a representation-theoretical significance: it is an eigenvalue equation for the Casimir operator of a group of space-time symmetries, picking out an irreducible representation of that group. In the non-relativistic case the Laplacian was the Casimir and the symmetry group was the Euclidean group E(3). In the relativistic case the Casimir operator is the Klein-Gordon operator, and the space-time symmetry group is the Poincare group. The Poincare group acts on the phase space of solutions to the Klein-Gordon equation, preserving the Poisson bracket. One can thus use the same methods as in the finite-dimensional case to get a representation of the Poincare group by intertwining operators for the Heisenberg Lie algebra representation (the representation given by the field operators). These methods give a representation of the Lie algebra of the Poincare group in terms of quadratic combinations of the field operators.

which views the K-G equation as coming from a Casimir, talks about how this Heisenberg Lie algebra can be used to find representations of the Poincare group and how one should form quadratic combinations of fields, motivated by thinking about intertwining operators...

My questions are -  based on the two outlines and Woit's quotes I've given above:

  1. What is wrong with the first outline I have given?

    e.g. Why is it wrong to take Minkowski space as the manifold, to ignore the classical Klein-Gordon equation, and to set up an eigenvalue equation where the eigenfunctions are Lie algebra operators?
  2. How do we clean up the second outline and make it look coherent?

    e.g. I have almost certainly misunderstood it, and not understood the link between the Poincare algebra coming from the Lie group and the Heisenberg Lie algebra coming from the dual of the space of solutions of K-G, let alone the link between the Poincare group and that Heisenberg Lie algebra, and why one can get the Poincare group from the Heisenberg Lie algebra.
  3. What is going on with this quadratic combinations ~ Intertwining operator ~ Hamiltonian business?

    e.g. it seems like this step is done to find a representation of the Poincare group even though you use representations of the dual, then you commutator this with those dual representations as if they're all in the same space anyway which is random, and even thinking to combine quadratic combinations comes out of nowhere. What's going on here?
  4. Can a simpler and clearer explanation of this process be given?

    e.g. In other words, what exactly is the process from start to finish?

References: Woit "Quantum Theory, Groups and Representations: An Introduction"

asked Aug 20, 2015 in Theoretical Physics by bolbteppa (120 points) [ no revision ]

The group you should use is the semidirect product of the Heisenberg-Weyl group for the fields and the Poincare group acting on the field coordinates, and the corresponding Lie algebra. The Heisenberg algebra then provides the linear fields, and the poicare generators can be expressed as quadratic expressions in these fields. The Klein-Gordon differential operator appears as Casimir.

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