[Paper Review] Group Fourier transform and the phase space path integral for finite dimensional Lie groups

Matti Raasakka|arXiv (Cornell University)|Nov 28, 2011

Mathematical Analysis and Transform Methods1 references3 citations

TL;DR

This paper introduces a group Fourier transform for finite-dimensional Lie groups, enabling a unitary mapping from functions on the group to a non-commutative dual space of momentum-like variables. It derives a first-order phase space path integral for quantum mechanics on such groups, with quantum-corrected Hamiltonians arising from non-commutative star products, offering a physically intuitive alternative to representation theory and special functions in quantum gravity models.

ABSTRACT

We formulate a notion of group Fourier transform for a finite dimensional Lie group. The transform provides a unitary map from square integrable functions on the group to square integrable functions on a non-commutative dual space. We then derive the first order phase space path integral for quantum mechanics on the group by using a non-commutative dual space representation obtained through the transform. Possible advantages of the formalism include: (1) The transform provides an alternative to the spectral decomposition via representation theory of Lie groups and the use of special functions. (2) The non-commutative dual variables are physically more intuitive, since despite the non-commutativity they are analogous to the corresponding classical variables. The work is expected, among other possible applications, to allow for the metric representation of Lorentzian spin foam models in the context of quantum gravity.

Motivation & Objective

• To generalize the group Fourier transform beyond SO(3) and SU(2) to arbitrary finite-dimensional Lie groups.
• To provide a non-commutative dual space representation for quantum mechanics on Lie groups, avoiding reliance on representation theory and special functions.
• To derive a first-order phase space path integral formulation using the dual variables, ensuring consistency with the Schrödinger equation.
• To enable a metric representation of quantum gravity models, particularly for Lorentzian spin foam models in 4D quantum gravity.
• To simplify the semi-classical analysis of quantum systems by replacing special functions with geometrically intuitive non-commutative variables.

Proposed method

• Define a non-commutative plane wave $ E_g(X) $ on the cotangent bundle $ \mathcal{G} \times \mathfrak{g}^* $, satisfying $ \delta_e(g) = \int_{\mathfrak{g}^*} \frac{dX}{(2\pi)^d} E_g(X) $.
• Introduce a $ \star $-product on functions on $ \mathfrak{g}^* $ such that $ E_g(X) \star E_h(X) = E_{gh}(X) $, reflecting the group structure.
• Define the group Fourier transform $ \tilde{\phi}(X) = \int_{\mathcal{G}} dg \, E_{g^{-1}}(X) \phi(g) $, mapping $ L^2(\mathcal{G}) $ to $ L^2_\star(\mathfrak{g}^*, dX) $.
• Establish unitarity of the transform under the condition $ E_{g^{-1}}(X) = \overline{E_g(X)} $, ensuring isometric mapping.
• Construct the phase space path integral via discretized time evolution, using $ \langle g_{k+1} | X_k \rangle \star \langle X_k | e^{-i\epsilon \hat{H}} | g_k \rangle $, with $ \hat{H} $ represented via $ H_\star(g,X) $.
• Take the continuum limit to obtain the path integral $ \langle g',t' | g,t \rangle = \int \mathcal{D}g \mathcal{D}X \, \exp\left\{ i \int_t^{t'} dt \, L_q(g,X) \right\} $, with quantum-corrected Lagrangian $ L_q = V \cdot X - H_q(g,X) $.

Experimental results

Research questions

• RQ1 How can a unitary group Fourier transform be consistently defined for an arbitrary finite-dimensional Lie group, generalizing previous work on SO(3) and SU(2)?
• RQ2 What is the structure of the non-commutative dual space $ \mathfrak{g}^* $, and how does it support a physically intuitive phase space formulation?
• RQ3 How can the first-order phase space path integral for quantum mechanics on a Lie group be derived using the group Fourier transform and $ \star $-product formalism?
• RQ4 What are the quantum corrections to the classical Hamiltonian and Lagrangian that arise from non-commutativity in the dual space?
• RQ5 Can this formalism simplify the semi-classical analysis and geometric interpretation of quantum gravity models such as Lorentzian spin foams?

Key findings

• The group Fourier transform provides a unitary isomorphism between $ L^2(\mathcal{G}) $ and $ L^2_\star(\mathfrak{g}^*, dX) $, with the $ \star $-product encoding the group structure.
• The inverse transform is given by $ \phi(g) = \int_{\mathfrak{g}^*} \frac{dX}{(2\pi)^d} \, E_g(X) \star \tilde{\phi}(X) $, ensuring completeness and isometry.
• The phase space path integral is derived as $ \langle g',t' | g,t \rangle = \lim_{\epsilon \to 0, N \to \infty} \prod_{k=0}^{N-1} \int_{\mathcal{G}} dg_k \int_{\mathfrak{g}^*} \frac{dX_k}{(2\pi)^d} \, \exp\left\{ i \sum_{k=0}^{N-1} \epsilon \left( \frac{1}{\epsilon} Z(g_k^{-1}g_{k+1}) \cdot X_k - H_q(g_k,X_k) \right) \right\} $.
• The quantum-corrected Hamiltonian $ H_q(g,X) = \omega^{-1}(-i\partial_X^i) H_\star(g,X) $ ensures the path integral satisfies the Schrödinger equation.
• In the continuum limit, the path integral takes the form $ \int \mathcal{D}g \mathcal{D}X \, \exp\left\{ i \int_t^{t'} dt \, (V \cdot X - H_q(g,X)) \right\} $, with $ V^i(t) = \lim_{\epsilon \to 0} Z^i(g^{-1}(t)g(t+\epsilon))/\epsilon $.
• For $ \mathcal{G} = \mathbb{R}^d $, the formalism reproduces the standard first-order path integral, validating its consistency.