Geometrical Evaluation of Star Products

The geometric picture of the star-product based on its Fourier representation kernel is utilized in the evaluation of chains of star-products and the intuitive appreciation of their associativity and symmetries. Such constructions appear even simpler for a variant asymmetric product, and carry through for the standard star-product supersymmetrization. §1. Groenewold’s noncommutative ⋆-product [1] of phase-space functions f(x, p) and g(x, p) is the unique associative pseudodifferential deformation [2] of ordinary products: ⋆ ≡ e ← ∂ x → ∂ p− ← ∂ p → ∂ x)/2 . (1) It is the linchpin of deformation (phase-space) quantization [3, 2], as well as applications of matrix models and non-commutative geometry ideas in M-physics [4]. In practice, since it involves exponentials of derivative operators, it may be evaluated through translation of function arguments, f(x, p) ⋆ g(x, p) = f(x+ ih̄ 2 → ∂ p, p− ih̄ 2 → ∂ x) g(x, p). (2) However, explicit evaluations of long strings of star-products in this language frequently appear intractable, unless the phase-space functions consist of exponentials or simple polynomials [5]. Baker [6] has utilized the more practical Fourier representation of this product as an integral kernel: f ⋆ g = 1 h̄π2 ∫ dpdpdxdx f(x, p) g(x, p) exp (−2i h̄ ( p(x − x) + p(x − x) + p(x− x) ) ) . (3) The cyclic determinantal expression multiplying −2i/h̄ in the exponent is twice the area of the phase-space triangle (r, r, r), where r ≡ (x, p), namely, 2A(r′′, r′, r) = (r′ − r) ∧ (r− r′′) = r′′ ∧ r′ + r′ ∧ r+ r ∧ r′′. (4)