Deformation Quantization and the Moyal Star Product Juan Montoya August 23, 2018 1 Deformation Quantization Quantization in general is a process in which one transfers from a system obeying classical mechanics into one following a quantum regime. Deformation quantization is a specific approach to this study, where a Poisson bracket is used to deform the function algebras on the commutative operators in a classical system. This allows for the construction of a new non-commutative product appropriate to describe a quantum system. One benefit of deformation quantization over other such quantization approaches is that the classical limit automatically functions intuitively, as the quantum system is derived from a classical system such as a Hamiltonian regime. Flato, Lichnerowicz and Sternheimer suggest that quantization be understood as a deformation of the structure of the algebra of classical observables rather than a radical change in the nature of the observables. In particular, deformation quantization allows the construction of an appropriate non-commutative product, that was shown not to exist in the case of canonical quantization, as there is no appropriate quantizing map Q from the set P ol(p ) of phase space polynomials in P to the set Op(H, D) of self adjoint operators in a Hilbert space H, both over a common invariant dense domain D such that Q : P ol(p ) Op(H, D), f Q(f) ˆf with the conditions that The constant function 1 maps to the unit operator: Linearity in Q: Q(1) ˆ1 Q(f + λg) Q(f) + λq(g) Compatibility of Lie structures on P ol(p ) and Op(H, D): Consistency with Schrödinger quantization: Q({f, g} P B ) 1 [Q(f), Q(g)] i (Q(q i )ψ)(q) q i ψ(q) and (Q(p i )ψ)(q) i qi ψ(q) Groenewold and van Howe showed that such a map Q does not exist. The star product is then introduced and constructed to circumvent this issue through the introduction of a noncommuting product, which emulates the non-commutativity of the quantum operators. So the star product must fulfill Q(f)Q(g) Q(f g) 1
It is important to note that different quantization schemes result in different star products. As the Weyl scheme is the most fundamental quantization scheme, this is the first to be considered for the construction of a star product, and the star product that corresponds to the Weyl scheme is known as the Moyal product, or the Moyal-Weyl product. For this short paper, the star product will refer to the Moyal product. 2 Preliminaries and Definitions A Lie algebra consists of a finite dimensional vector space V over a field F, and a Lie bracket (multiplication) {, } : V V V is a bilinear map with the properties {v, v} 0 {u, {v, w}} + {v, {u, w}} + {w, {u, v}} 0 For u, v, w V. {u, v} {v, u} follows from bilinearity in our Lie algebra. A Poisson algebra is defined as the real valued vector space A with a commutative associative algebra structure (f, g) fg and a Lie algebra structure such that (f, g) {f, g} which together satisfy the compatability condition {fg, h} f{g, h} + {f, h}g A Poisson manifold, denoted as (M, P ), is a manifold with the function space C (M) with regard to smooth functions on the manifold M, with an associated Poisson tensor over the Poisson Manifold. The Poisson tensor is defined as {u, v} P (du dv) n u v u v p i q i q i p i i1 This Poisson tensor is used to define the Poisson bracket, which is an R bilinear map on C (M), (u, v) {u, v} with the following properties for u, v, w C (M): {u, v} {v, u} 0 {{u, v}, w} + {{v, w}, u} + {{w, u}, v} {u, vw} {u, v}w + {u, w}v The duals of Lie algebras also form linear Poisson manifolds. If g is a Lie algebra, then its 2
dual g has a Poisson tensor P defined by P ξ (X, Y ) : ξ([x, Y ]) For X, Y g (g ) A star product on a Poisson manifold (M, P ) is a biliniear map N N N[v], (u, v) u v u ν v : r 0 ν r C r (u, v) Where N C (M), such that is formally associative, with (u v) w u (v w) C 0 (u, v) uv, C 1 (u, v) C 1 (v, u) {u, v} 1 u u 1 u When each C r is a bidifferential opperator on M, the resulting star product is a differential star product, and when each C r is of order maximum r in each arguement, one considers the natural star product. In addition, a star product may also be defined on a subspace N C (M) where N is stable under pointwise multiplication and a Poisson bracket. A Poisson deformation of the Poisson Bracket on a Poisson manifold (M, P ) is a formal deformation {, } ν of the Lie algebra (C (M), {, }) so that for all u (C (M), {u, } ν P ν (du, dv) where P ν P + ν k P k is a series of skew symmetric contravariant two-tensors on M such that [P ν, P ν ] 0. 3 The Weyl scheme and Moyal product This transition from a classical Lie structure into a Poisson manifold with a non-commuting Poisson bracket is the deformation on the system to give rise to quantum behavior in a system. We will next consider the the Weyl scheme. Weyl showed that a quantizing map Q W (f(p, q)) can be represented through taking the Fourier transform of f, and then creating an operator out of the Fourier transform back into f f(u, v) 1 () 2 f(p, q)e i(uq+vp) dpdq Q W (f(q, p)) f(u, v)e i(uˆq+v ˆp) dudv Using the aforementioned requirement for the star product Q(f)Q(g) Q(f g), we can then write the quantization of a star product of functions f, g as 3
g(u u 1, v v 1 )du 1 dv 1 Q W (f g) Q W (f)q W (g) f(u 1, v 1 ) g(u 2, v 2 )e i(u 1 ˆq+v 1 ˆp) e i(u 2 ˆq+v 2 ˆp) du 1 dv 1 du 2 dv 2 f(u 1, v 1 ) g(u 2, v 2 )e i((u 1+u 2 )ˆq+(v 1 +v 2 )ˆp) e i 2 (u 1v 2 v 1 u 2 ) du 1 dv 1 du 2 dv 2 e i(uˆq+v ˆp) dudv m,n0 ( 1) m ( i ) m+nu m m!n! 2 1 v1 n f(u 1, v 1 )(u u 1 ) n (v v 1 ) m The middle line is acquired using the the truncated Cambell-Baker-Hausdorff formula eâe ˆB e (Â+ ˆB) e 1 2 [Â, ˆB], and the last line is acquired using the subsitution v v 1 +v 2, u u 1 +u 2. The latter integral in the final line is the Fourier transform of the expression for the Moyal product (by Fourier convolution theorem), thus: Q W (f)q W (g) e i(uˆq+v ˆp) ( f g)dudv So the Moyal product on two functions can then be written as Where (f g)(q, p) m,n0 n i1 ( 1) m ( i ) m+n( m m!n! 2 p q n f)(p n q m g) ( i ) n ( p i1 ( i P ) n (f g) n0 exp (i P )(f g) P q i1 q i1 n i0 p i1 )(... )( p in p i q i q i p i q in q in p in )(f g) The Moyal product (or more generally for other quantum regimes, the star product) replaces the quantizing operator Q, which allows the new structure to properly exhibit the noncommutative behavior of a quantum regime, and it is this application of the star product that deforms the Lie algebra. The Lie algebra in isolation behaves as a classical system would; the transition into a quantum regime is introduced with the fourier transform and convolution to generate a the appropriate phase space variables. The Poisson algebra and its corresponding Poisson bracket are the manifestations of quantum phenomena out of a classical system. In order to apply the Moyal product, it is also necessary to find appropriate phase space functions that correspond to the quantum mechanical states. These states may be constructed through the inverse of the quantization map, which is given in general by Q 1 φ ( ˆf(ˆq, ˆp)) sπ T r[ ˆf(ˆq, ˆp) φ 1 µ,νλ ( u, v)ei(u(ˆq q)+v(ˆp p)) dudv] 4
In the case of the Weyl regime, these quantum states can be found through the specific map Q 1 W ( ˆf(ˆq, ˆp)) dudve i(uq+vp) T r[ ˆf(ˆq, ˆp)e i(uˆq+v ˆp) ] dudve i(uq+vp) e i 2 uv T r[ dq q q ˆf(ˆq, ˆp)e iv ˆp e iuˆq ] dudve i(uq+vp) e i 2 uv dq q ˆf(ˆq, ˆp)e iv ˆp q e iuq dudvdq e iu(q q 2 v) q ˆf(ˆq, ˆp) q v de ivp dv q + 2 v ˆf(ˆq, ˆp) q 2 v e ivp With this Q 1, we can find the phase space analogue and find the phase space function corresponding to the density matrix ˆp ψ ψ, which in the case of the Weyl regime gives the Wigner function π(q, p) dv ψ(q + 2 )ψ(q 2 )e ipv Now, with the Wigner functions, and the Hamilton operator Ĥ, we can find through the same methods that the quantization of our Hamilton operator Q 1 W (Ĥ ˆρ) H π(q, p), which from there we can derive that H π E E π E This is the phase space equivalent of Ĥ ˆρ E ˆρ. calculated, where we find that π(q, p; t) U(t) π H (q, p Ū)t The time density matrix can also be Where i t π(q, p; t) [H(q, p; t), π(q, p; t)], and π(q, p; t) is the Wigner function in the Schrödinger picture, and π H (q, p) π(q, p; t 0). All of these same methods can be applied to other quantum regimes, and would result in a different star product and results. But the Weyl regime and the Moyal star product were the conceptual and historical starting points for deformation quantization, in which quantum observables were first treated as a non-commuting product, rather than as an operator. This lends to the suggestion that rather than considering operators on a Hilbert space, it is possible to forget about the operator formalism and uniquely consider the phase space for the quantum system. The observables of the system are functions on the phase space, and the states of the system are positive functionals on the observables (in this case the Dirac function), and one obtains the value of the observable in the definite state through evaluating the star product on the phase space functions. 5
References [1] Maciej B laszak and Ziemowit Domański. Phase space quantum mechanics. Annals of Physics, 327(2):167 211, 2012. [2] Simone Gutt. Deformation Quantization: an introduction. PhD thesis, Quantifization and Harmonic Analysis, 2005. [3] Peter Henselder, Allen C Hirshfeld, and Thomas Spernat. Star products and geometric algebra. Annals of Physics, 317(1):107 129, 2005. [4] Gizem Karaali. Deformation quantization-a brief survey, 2016. 6