hara@math.kyushu-u.ac.jp 1 1 1.1............................................... 2 1.2............................................. 3 2 3 3 5 3.1............................................. 6 3.2................................... 10 3.3........................................ 12 3.4................................................. 13 3.5......................................... 17 4 18 4.1.......................................... 19 4.2.................................... 20 4.3................................................. 20 5 21 5.1.............................. 21 5.2................................................. 21 5.3.......................................... 22 6 23 1 1
Hilbert, von Neuman 4 1. 2. 5.3 1.1 19 1 10 11 [1, p.86] kt 2 1 [1, 2] 2 2
1.1 ( ) 1.2 n 2 3 [1, 3] 2 3 3
5.3 4 5 q(t) t t = 0 q(0) dq (0) dt 6 7 q i (t) p i (t) i = 1, 2,..., n n n- (q 1 (t), q 2 (t),..., q n (t)) q(t) (p 1 (t), p 2 (t),..., p n (t)) p(t) (q(t), p(t)) 2.1 ( ) n- q i (t) p i (t) i = 1, 2,..., n (q(t), p(t)) i = 1, 2,..., n dp i dt = H(q, p) q i, dq i dt = H(q, p) p i (2.1) H(q, p) 4 5 6 3.1 7 4
(1) (2) V (q) V (q) m > 0 V (q) ω > 0 H(q, p) = p2 + V (q) (2.2) 2m V (q) = mω2 2 q2 (2.3) 3 von Neuman 8 5.3 8 3.4 [4] 5
3.5 9 3.1 3.1 3.3 3.4 3.1 3.1 3.4 3.5 3.1 ( I ) H 1 state 3.2 (Dirac I) Dirac [5] 1. H ψ φ 2. H H H ψ ψ 3. ψ φ ψ, φ ψ φ 3.3 ( 3.1 ) 3.1 3.1 ψ e iα ψ α R ray ψ ψ 3.1 10 3.3 3.5 3.4 ( II ) (a) observable H (b) I 9 10 3.4.6 6
operator (a) 3.5 (b) 1.1 (2.2) q p q p 3.2 qp ˆq, ˆp (ˆqˆp + ˆpˆq)/2 ψ Â 11 3.5 ( III ) (a) ψ Â Â (b) Â Â = d ˆP A (a) a (3.1) (a 1, a 2 ] ψ, { ˆPA (a 2 ) ˆP A (a 1 ) } ψ (3.2) 3.5 3.5 3.6 ( I) ψ Â ψ 3.5 ψ ψ ψ 3.5 ψ 3.5 12 3.5 11 3.22 12 7
ψ 3.5 II 3.7 3.5  ψ ψ  ψ 3.6 ψ  ψ = ψ, d ˆP A (a)ψ a (3.3) (q, p) {a n } n 13 ψ a n ψ, ˆP n ψ 3.1 3.4 3.5 3.1.1 Dirac Dirac Dirac  = ˆP n a n + d ˆP (a) a (3.4) n ˆP n  a n ˆP (a)  a 11 11 = n ˆP n + d ˆP (a) (3.5) ψ ψ = n ˆP n ψ + d ˆP (a) ψ (3.6) ψ  3.5 a n ψ, ˆP n ψ a ψ, d ˆP (a)ψ 13 8
Dirac Dirac 3.8 (Dirac II) Dirac 3.2 1. ϕ ˆP ϕ ˆP ϕ = ϕ ϕ ϕ ϕ ψ ϕ - ˆP ϕ ψ = ϕ ϕ ψ = ϕ ψ ϕ 2.  ϕ n a n ϕ n ϕ n ϕ n ϕ n a n a n ϕ n,m m ˆP n m ϕ n,m ϕ n,m 3. d ˆP (a) da ϕ(a) ϕ(a) d ˆP (a) da ϕ(a) ϕ(a) 4. ϕ(a) a n ϕ(a) a  a a, k k a 3.9 Dirac a (3.4) (3.6)  = a n a n a n + da a a a (3.7) n 11 = n ψ = n a n a n + a n ψ a n + da a a (3.8) da a ψ a (3.9) (3.9) a n ψ a ψ (3.7) (3.8) ψ  3.5 a n a n ψ 2 14 a a ψ 2 3.9 (Gelfand ) Dirac Dirac a a H H H Gelfand S(R n ) H = L 2 (R n ) S (R n ) [6] 14 k a n, k ψ 2 9
3.2 3.1 H H H 5.1 [7, 16.5] Â ˆB commutator [Â, ˆB] := Â ˆB ˆBÂ (3.10) q p 2 3.16 3.4 3.10 ( CCR) n- 15 ˆq j ˆp j Canonical Commutation Relation CCR [ˆq j, ˆp k ] = i δ j,k (3.11) [ˆq j, ˆq k ] = [ˆp j, ˆp k ] = 0 (3.12) j = 1, 2,..., n k = 1, 2,..., n 16 1.0545887 10 34 J sec (3.13) i i 11 i δ j,k { 1 (j = k) δ j,k = 0 (j k) CCR 3.11 ( ) Poisson {, } [, ] {A, B} = 1 [Â, ˆB] i q j p k Poisson δ j,k 3.11 3.10 (3.14) trace ˆq j ˆp j 17 ˆq, ˆp H 15 H 16 2π 17 ˆq, ˆp CCR ˆq n ˆp ˆpˆq n = in n ˆq n 1 n ˆq n 1 n 2 ˆq n ˆp ˆq n ˆq n 1 ˆq n n 2 ˆq ˆp n [7, p.318] 10
CCR H CCR CCR 3.12 ( Schrödinger ) H L 2 (R n ) j = 1, 2,..., n ˆq j : ψ(q) q j ψ(q) (3.15) {ψ(q) L 2 (R n ) : q j ψ(q) L 2 (R n )} (3.16) ˆp j : ψ(q) i ψ(q) q j (3.17) {ψ(q) L 2 (R n ) : ψ(q) ψ(q) q j L 2 (R n )} (3.18) CCR Schrödinger (Schrödinger representation) 18 CCR H S(R n ) ˆq j ˆp j CCR Schrödinger j CCR H 3.13 (Rellich-Dixmier) H ˆq, ˆp 1. H 2. ˆq, ˆp H Ω [ˆq, ˆp] = i 3. Ω ˆq 2 + ˆp 2 ˆq, ˆp CCR Schrödinger Schrödinger CCR Weyl 3.14 ( Weyl ) Û(a) ˆV (b) a, b R Weyl Û(a) ˆV (b) = ˆV (b)û(a)e iab Û(a)Û(b) = Û(a + b), ˆV (a) ˆV (b) = ˆV (a + b) (3.19) 18 Dirac 3.8 ˆq q ψ(q) q ψ 11
a, b CCR Weyl Û(a) ˆV (b) ˆq ˆp Û(a) := e iaˆq, ˆV (b) := e ibˆp (3.20) CCR (3.19) H 3.15 (von Neuman) Hilbert CCR Weyl CCR Schrödinger CCR Schrödinger CCR Schrödinger 3.16 r p r CCR CCR Schrödinger (, ) r 0 t H H 3.3 3.3.1 ψ Â Â ψ, Âψ (3.21) 3.17 ( ) Ô ψ Ô Ô (Ô Ô ) 2 1/2 ψ ψ ψ (3.22) Â, ˆB ψ Â ˆB 1 2 [Â, ˆB] (3.23) ψ ˆq ˆp Schwartz t R ˆq j ˆp k 2 δ j,k (3.24) φ ( Â + it ˆB ) ψ (3.25) ( ) 2 Â ˆB Â Â ψ ˆB ˆB ψ 12
3.3.2 Â Â 3.6 3.17 ψ Â ˆB (3.24) Â ˆB ψ Â ˆB 3.6 Â ˆB 3.5 3.3.3 3.1 3.1 (3.13) (3.24) 1.2 k p p = k 3.1 3.1 3.1 Heisenberg [8] 3.4 3.1 3.3 3.4.7 13
Schrödinger picture Heisenberg picture interaction picture Schrödinger picture Schrödinger picture Schrödinger representation 3.4.1 Schrödinger picture 3.18 ( Schrödinger picture) Ĥ Schrödinger t ψ(t) i d ψ(t) = Ĥ ψ(t) (3.26) dt Ĥ 3.4 ˆq ˆp 3.19 ( ) 1. picture CCR representation 2. Schrödinger (3.26) (3.26) Schrödinger H 19 (3.26) (3.26) ) (Ĥt ψ(t) = exp ψ(0) (3.27) i dp H (Ĥt ) exp i exp ( ) Et dp H (E) (3.28) i (3.26) H (3.27) 3.4.2 pictures Schrödinger picture Heisenberg Picture Interaction picture 3.1 Schrödinger picture Schrödinger picture 19 (2.2) ˆp2 2 Schrödinger 3.12 14
3.20 ( Heisenberg picture) i d dtâh(t) = [ÂH Ĥ] (t), (3.29) ( ) (Ĥt )  H (t) = exp Ĥt  H (0) exp i i (3.29) (3.30) Schrödinger picture Heisenberg picture 3.21 ( ) (3.30) (3.29) d A(t) = {A(t), H} (3.31) dt {, } Poisson 3.11 3.20 3.11 3.11 (3.31) 3.20 Heisenberg picture Schrödinger picture 3.18 3.11 Interaction Picture Ĥ = Ĥ0 + Ĥint (3.32) Ĥ0 Ĥ0 Ĥint eĥt/(i ) (Ĥt ) ) ) (Ĥ0 t (Ĥint t exp = exp exp i i i ( ) ) (Ĥ0  int (t) = exp Ĥ0t t  int (0) exp i i ψ(t) int = exp (Ĥint t i (3.33) (3.34) ) ψ(0) int (3.35) Ĥ0 Ĥint (3.34) i d dt ψ(t) = Ĥint(t) ψ(t) (3.36) Ĥint(t) (3.34) (3.36) chronological exponential 20 [ ( ) n 1 t tn ] t2 ψ(t) int = dt n dt n 1 dt 1 Ĥ int (t n )Ĥint(t n 1 ) i Ĥint(t 1 ) ψ(0) int 0 0 0 n=0 20 (3.37) 15
{ 1 T-exp i t 0 } dsĥint(s) 3.4.3 (3.27) ψ(t) ψ(t) = ψ(0) ψ(0) = 1 (3.38) 3.1 t = 0 t > 0 3.4.4 Schrödinger picture ψ Ĥ E Ĥ ψ = E ψ (3.39) ψ(t) = e iet/ ψ (3.40) 3.3 (3.39) Schrödinger 3.4.5  Ĥ [Â, Ĥ ] = 0 (3.41)  1. ψ a  a  2. ψ   1  a 2  ψ t = 0 ψ t = 0  t = 1  t = 0 t = 1  3.5 16
3.4.6 3.1 0 1 21 3.4.7 (3.26) (3.29) Ĥ chronological exponential t = 0 t > 0 3.5 3.5 3.22 ( IV II) ψ   3.5 a n ϕ n ψ  a n ϕ n  3.5 a n 3.5 ψ ψ  ψ 3.5 1920 3.5.1 21 17
3.22 1. 3.4 3.18 (3.27) (3.27) 2. 3.22 ψ Â Â 22 Â 3.5 (3.27) 23 3.18 3.22 24 von Neuman [9, ] Pauli [10] 4 22 4.1.1 23 24 18
4.1 V (q) m Ĥ = ˆp2 + V (q) (4.1) 2m Schrödinger ψ(q, t) = φ(q)e iet/ ψ(q, t) i = 2 2 ψ(q, t) t 2m q 2 + V (q)ψ(q, t) (4.2) Eφ(q) = 2 d 2 φ(q) 2m dq 2 + V (q)φ(q) (4.3) L 2 (R) V (q) V (q) Sturm Liouville [1, 3, 5, 11, 12] 4.1.1 V (q) 0 (4.2) 3.3 t = 0 q = 0 q = 0 Dirac H = L 2 (R) q = 0 ψ(q, 0) π 2 α e q /α (4.4) 0 < α 1 0 q = 0 (, ) 4.1.2 V (q) = mω2 2 q2 n = 0, 1, 2,... Eφ(q) = 2 d 2 φ(q) 2m dq 2 + mω2 2 q2 φ(q) (4.5) E n = ( n + 1 ) ω (4.6) 2 φ n (q) C n H n (x)e x2 /4, x=(2mω/ ) 1/2 q ( ) 1/2 1 (mω ) 1/4 C n (4.7) n! π 19
H n (x) ( 1) n e x2 /2 d n dx e x2 /2 n n {φ n } n H = L 2 (R) 4.2 Ĥ = 2 2m 2 Q2 r (4.8) 2 r Q Q Q E < 0 E 0 E < 0 E n = mq4 2 2 1 n 2 (n = 1, 2,...) (4.9) n E n n 2 E > 0 4.1.2 n E n E m = mq4 2 2 ( 1 n 2 1 ) m 2 (4.10) 25 ( 1 n 2 1 m 2 ) 4.3 [4, 13] [1, 3, 5, 11, 12] 25 Dirac 1920 20
5 5.1 3.1 [5, 25, 35] R ˆR ψ 1 = ˆR ψ 0 (5.1) R ˆR ˆR R R 1 R 2 R 2 R 1 ψ 1 ˆR 1 ( ψ 0 ) (5.2) ψ 0 1 2 = ˆR 2 ( ψ 1 ) = ˆR 2 ˆR 1 ( ψ 0 ) (5.3) ˆR 2 0 1 R 2 R 1 ˆR 2 ˆR 1 H space H spin H total = H space H spin (5.4) H space 3.2 H spin 2n + 1 n 1/2 1/2 SO(3) SU(2) O(3) SU(2) 5.2 3.2 canonical quantization Feynman path integral 1. 21
2. 3. 4. [14, 13] 5.3 5.3.1 5.1 ( ) 1. 2. Dirac 1.2 5.3.2 4 5.3.1 Dirac, Heisenberg Dirac 1.2 22
5.3.3 CCR 26 5.3.2 CPT C - von Neuman 6 [9] [7] [15] 1970 [16] [5, 1, 3, 2, 17] [5] [9, 7] [1, 3, 2] [1, 3] 26 1960 Fock 60 Higgs Weinberg-Salam 2008 23
[17] 5.1 [11, 12] [6, 18, 19, 20] [1]. I., (1977(2e)). [2].., (1977). [3]. II., (1952). [4].., pp. 669 686., (1978). [5] P.A.M. Dirac. The Principles of Quantum Mechanics. Oxford, (1958). [6] Bogoliubov, Lognov, and Todorov. Axiomatic Quantum Field Theory., (1980).. [7].. II, 4, pp. 247 484., (1978). [8].. II, 4, pp. 557 602., (1978). [9] von Neuman. Die Mathematische Grundlagen der Quantenmechanik. Springer, (1932). 1957. [10].., pp. 557 594., (1978). [11] L.D. Landau and I.M. Lifshitz. I II.., (19 ). [12],,,,,,. I. 3., (1978). [13] B. Simon. Functional Integration and Quantum Physics. Academic Press, (1979). [14] R.P. Feynman and Hibbs. Path Integrals and Quantum Mechanics. MacGrow-Hill, (1965). [15]... [16],.., (1978). [17] R.P. Feynman, R.B. Leighton, and M. Sands. Quantum Mechanics. The Feynman Lectures on Physics. Addison-Wesley, (1965). [18] R.F. Streater and A.S. Wightman. PCT, Spin and Statistics, and All That. Benjamin, (1964). [19] R. Fernández, J. Fröhlich, and A.D. Sokal. Random Walks, Critical Phenomena, and Tiviality in Quantum Field Theory. Springer, (1992). [20] O. Brattelli and H. Robinson. C -algebras and Quantum Statistical Mechanics. Springer, (19 ). 24