30 3..........................................................................................3.................................... 4.4..................................... 6 A Q, P s- 7 B α- 9 Q P () ρ W (q, p) def dkdχ (π) e i(kq+χp) Tr[e i(kq+χp ) ρ] Tr[Π(q, p)ρ] (.) Π(q, p) def dkdχ (π) e i(kq+χp) e i(kq+χp ). (.) f(q, p) Q[f(q, p)] Q[f(q, p)] def dqdp f(q, p)π(q, p) (.3) Π(q, p) f Q[f(q, p)] Q[f(q, p)] ρ Q[f(q, p)] Tr(Q[f(q, p)]ρ) dqdp f(q, p)w (q, p) (.4)

Q[{f, g}] [Q[f], Q[g]] iħ + O(ħ) (.5) [4] {f, g} [F, G] Q[q n p m ] l0 m m C l P l Q n P m l n k0 n n C k Q k P m Q n k. (.6).4 Q[qp] (QP + P Q). (.7) [, 3]. A, B W (a, b) def dsdt (π) e i(sa+tb) Tr[e i(sa+tb) ρ] Tr[Π(a, b)ρ], (.) Π(a, b) def dsdt (π) e i(sa+tb) e i(sa+tb) (.) f(a, b) Q[f(q, p)] Q[f(a, b)] def dadb f(a, b)π(a, b) (.3) Q[f(a, b)] Tr(Q[f(a, b)]ρ) dadb f(a, b)w (q, p) (.4) n [, 4].. #(s, t) A, B A, B c a, b e i(sa+tb) #(s, 0) e isa, (.5) #(0, t) e itb (.6)

#(s, t) e i(sa+tb) e isa e itb e itb e isa Π n k eiαksa eiβktb ( n k α k, n k β k ) (.7) #(s, t) α e isa e itb + + α e itb e isa (.8) #(s, t) dk G(k)# k (s, t) ( ) dk G(k), (.9) # k (s, t) def e i s ( k)a e itb e i s (+k)a (.0) #(s, t) e isa e itb Kirkwood(933)-Dirac(945) [5, 6]. #(s, t) e isa e itb 949 Moyal [7]. (A, B) (Q, P ) Weyl(97) Wigner(93) [8, 9]. W # (a, b) def dsdt (π) e i(sa+tb) Tr[#(s, t)ρ] Tr[Π # (a, b)ρ], (.) Π # (a, b) def dsdt (π) e i(sa+tb) #(s, t) (.) f(a, b) Q # [f(q, p)] Q # [f(a, b)] def dadb f(a, b)π # (a, b) (.3) Q # [f(a, b)] Tr(Q # [f(a, b)]ρ) dadb f(a, b)w # (q, p) (.4) W # (a, b) a( b) b( a) dsdt db Π # (a, b) (π) db e i(sa+tb) #(s, t) dsdt π e isa δ(t)#(s, t) ds π e isa #(s, 0) ds π eis(a a) δ(a a) (.5) 3

db W # (a, b) Tr[δ(A a)ρ] W (a) (.6) a da W # (a, b) Tr[δ(B b)ρ] W (b). (.7) Q # [f(a)] f(a), (.8) Q # [g(b)] g(b) (.9) Q # [a n b m ] Q # [a n b m ] dadb a n b m Π # (a, b) dadb dadb dsdt (π) an b m e i(sa+tb) #(s, t) dsdt (π) dsdt (π) n+m #(s, t) (is) n (it) m n+m e i(sa+tb) ( is) n #(s, t) ( it) m dadb e i(sa+tb) n+m #(s, t) (is) n (it) m s0,t0 (.0).3 Weyl and Wigner[8, 9] (a n b m ) WW l0 m m C l B l A n B m l (.) Kirkwood and Dirac[5, 6] (a n b m ) KD A n B m, B m A n (.) Margenau and Hill[0] (a n b m ) MH (An B m + B m A n ) (.3) Born and Jordan[] (a n b m ) BJ m + 4 B l A n B m l. (.4) l0

Q[X] (X) (.9) #(s, t) (.0) (a n b m ) # G m,l def l0 l0 m m C l dk G(k)( + k) l ( k) m l B l A n B m l m m C l G m,l B l A n B m l, (.5) dk G(k)( + k) l ( k) m l (.6) Weyl and Wigner G(k) δ(k) G m,l, (.7) (a n b m ) # m m C l B l A n B m l (.8) Margenau and Hill G(k) [δ(k ) + δ(k )] l0 Born and Jordan G m,l m (δ l0 + δ lm ), (.9) (a n b m ) # (An B m + B m A n ) (.30) G(k) { x 0 x > (.3) ) G m,l def dk ( + k)l ( k) m l m dk x l ( x) m l 0 m B(l +, m l + ) m Γ(l + )Γ(m l + ) Γ(m + ) m l!(m l)! (m + )m! m, m + mc l (.3) (a n b m ) # B l A n B m l. m + (.33) l0 ) G(k) sin x/x 5

.4 (A, B) (Q, P ) g(x) g(0) (.34) #(s, t) g( ħst )# 0(s, t) (.35) # 0 (s, t) e i(sq+tp ) (.36) g(x), cos x, sin x x Weyl and Wigner, Margenau and Hill, Born and Jordan g(x) g(x) dk G(k)e ikx. (.37) (.34) dk G(k). (.38) #(s, t) (.9) #(s, t) ħst ik dk G(k)e #0 (s, t) dk G(k)# k (s, t), # k (s, t) def ħst ik e #0 (s, t) (.39) # k (s, t) e i k sq # 0 (s, t)e i k sq e i k tp # 0 (s, t)e i k tp (.40) # 0 (s, t) e isq/ e itp e isq/ e itp/ e isq e itp/ (.4) # k (s, t) Q, P (.) (.6) # k (s, t) e i s ( k)q e itp e i s (+k)q (.4) e i t (+k)p e isq e i t ( k)p (.43) 6

A Q, P s- [4] a, a ) [a, a ], [a, a] 0 [a, a ] (A.) D(α) def exp(αa α a), (A.) D(α, s) def D(α) exp( s α ), (A.3) s (z) def d α π D(α, s) exp( αz + α z) (A.4) a, a ρ ρ 3) α D(α) 0 (a 0 0, 0 0 ) (A.6) ρ s (z) def Tr[ρ s (z)] (A.7) d α +α z π e αz e s α Tr[e αa α a ρ] (A.8) ρ d z π ρ s(z) s (z) (A.9) a, a A d z A π A s(z) s (z), (A.0) A s (z) def Tr[A s (z)] (A.) Tr[AB] d z d π A z s(z)b s (z) π A s(z)b s (z) (A.) d α A π Tr[AD (α)]d(α) (A.3) ) α x + iy re iθ d α dx dy 0 dr π 0 dθ r (A.5) 3) (A, B) (Q, P ) 7

s,, 0 P, Q, Wigner (A.8) W (z) ρ 0 (z) d α π e αz +α z Tr[e αa α a ρ] (A.4) (z) z z (A.5) (A.9) ρ d z π P (z) z z, P (z) ρ (z) (A.6) (A.7) Q(z) ρ (z) z ρ z (A.7) s- {(a ) n a m } s def ( ) m n+m α n α m D(α, s) α0 (A.8) d z π z n z m s (z) {(a ) n a m } s, (A.9) (A.0) ({(a ) n a m } s ) s (z) z n z m (A.) (A.) {(a ) n a m } s d z π ρ s(z)z n z m (A.) D(α, s) D(α)e s α e s α e αa e α a e s e α a e αa (A.3) n+m {(a ) n a m } ( ) m α n α m e α a e αa α0 a m (a ) n, (A.4) n+m {(a ) n a m } ( ) m α n α m eαa e α a α0 (a ) n a m (A.5) 8

(a ) n a m a m (a ) n d z π P (z)z n z m, (A.6) d z π Q(z)z n z m (A.7) n+m {(a ) n a m } 0 ( ) m α a α n α m eαa α0 n+m ( )m k! α n α m (αa α a) k α0 k0 n+m ( )m (n + m)! α n α m (αa α a) n+m α0 n+m (n + m)! α n α m (αa + α a) n+m α0 (A.8) {a a} 0 a a + aa {(a ) n a m } 0 a, a Weyl (A.9) B α- A α (x, p) def dq e ipq x + α q A x + α q (B.) α 0, A ρ Wigner [4]. A α (x) def dp π A α(x, p) dp dq e ipq x + α q A x + α q π x A x (B.) A α (p) def dp (π) dx (π) A α(x, p) dx dq e ipq x + α q A x + α q (B.3) x x + α q, x x + α q (B.4) x x + x + α q, q x x (B.5) 9

( < α ) A α (p) (π) dx dx dx e ip(x x ) x A x dx p x x A x x p p A p (B.6) A α dxdp π dx dx α (x, p) def dq A α dq x α def dq e iqp x + + α dy e ip(q+y) x + + α q x α q (B.7) dxdp π A α(x, p) α (x, p) (B.8) q x α dx x x A x x y x α q A x + + α q A x + + α q x + + α q q x α y A A (B.9) dxdp π A α(x, p) α (x, p) (B.0) A α (x, p) Tr[A α (x, p)] Tr[A α(x, p)] (B.) A E a,b α (x, p) E a,b α (x, p) dxdp π Tr[A α(x, p)] α (x, p). (B.) A E a,b def e i(ax+bp ) (B.3) dq e ipq x + α dq e ipq e iab/ x + α dq e ipq e iab/ +α ia(x e q e i(ax+bp ) x + α q (B.4) q e iax e ibp x + α q q x + α q b q) x + α dq e ipq e iab/ +α ia(x e q) δ(q b) e ipb e iab/ +α ia(x e b) e i(ax+bp αab/) (B.5) 0

α, 0, (e iax e ibp ) (x, p) (e i(ax+bp ) ) 0 (x, p) (e ibp e iax ) (x, p) e iax+ibp (B.6) α- {X n P m } α def dxdp π xn p m α (x, p) (B.7) dxdp π eiax e ibp (x, p) e iax e ibp, (B.8) dxdp π eiax e ibp 0 (x, p) e i(ax+bp ), (B.9) dxdp π eiax e ibp (x, p) e ibp e iax (B.0) {X n P m } X n P m, {X n P m } P m X n (B.) {X n P m } 0 n+m (ax + bp ) n+m a,b0 (n + m)! a n b m (B.) [], 5, pan. [],, Quasi-probabilities in conditioned quantum measurement and a geometric/statistical interpretation of Aharonov s weak value, PTEP 07.5 (07): 05A0 [3],, On Quantisations, Quasi-probabilities and the Weak Value, arxiv:703.06068 [4] c [5] J. G. Kirkwood, Quantum Statistics of Almost Classical Assemblies, Phys. Rev. 44, 3 (933). [6] P. A. M. Dirac, On the Analogy Between Classical and Quantum Mechanics, Rev. Mod. Phys. 7, 95 (945). [7] J. E. Moyal, Math. Proc. Cambridge Philos. Soc. 45, 99 (949). [8] H. Weyl, Quantenmechanik und Gruppentheorie, Zeitschrift f ur Physik, 46(), (97). [9] E. Wigner, On the Quantum Correction For Thermodynamic Equilibrium, Phys. Rev. 40, 749 (93). [0] H. Margenau and N. R. Hill, Correlations between measurements in quantum theory, Prog. Theoret. Phys. 6, 7 (96). [] M. Born and P. Jordan, Zur Quantenmechanik, Z. Phys. 34, 858 (95).