\\(2 \\times 2\\) \u884c\u5217 \\(P =\\left( \\begin{array}{cc} p & q \\\\ r & s \\end{array} \\right)\\) \u306b\u5bfe\u3057\u3066\r\n\\[\r\n\\text{Tr}(P) =p+s\r\n\\]\r\n\u3068\u5b9a\u3081\u308b.
\r\n\u3000\\(a , b , c\\) \u306f \\(a \\geqq b >0\\) , \\(0 \\leqq c \\leqq 1\\) \u3092\u6e80\u305f\u3059\u5b9f\u6570\u3068\u3059\u308b. \u884c\u5217 \\(A , B , C , D\\) \u3092\u6b21\u3067\u5b9a\u3081\u308b.\r\n\\[\\begin{align}\r\nA & = \\left( \\begin{array}{cc} a & 0 \\\\ 0 & b \\end{array} \\right) , \\quad B =\\left( \\begin{array}{cc} b & 0 \\\\ 0 & a \\end{array} \\right) , \\\\\r\nC & =\\left( \\begin{array}{cc} a^c & 0 \\\\ 0 & b^c \\end{array} \\right) , \\quad D = \\left( \\begin{array}{cc} b^{1-c} & 0 \\\\ 0 & a^{1-c} \\end{array} \\right)\r\n\\end{align}\\]\r\n\u307e\u305f\u5b9f\u6570 \\(x\\) \u306b\u5bfe\u3057 \\(U(x) =\\left( \\begin{array}{cc} \\cos x & -\\sin x \\\\ \\sin x & \\cos x \\end{array} \\right)\\) \u3068\u3059\u308b.
\r\n\u3000\u3053\u306e\u3068\u304d\u4ee5\u4e0b\u306e\u554f\u3044\u306b\u7b54\u3048\u3088.<\/p>\r\n
(1)<\/strong>\u3000\u5404\u5b9f\u6570 \\(t\\) \u306b\u5bfe\u3057\u3066, \\(x\\) \u306e\u95a2\u6570\r\n\\[\r\nf(x) = \\text{Tr} \\left( \\left( U(t) A U(-t) -B \\right) U(x) \\left( \\begin{array}{cc} 1 & 0 \\\\ 0 & -1 \\end{array} \\right) U(-x) \\right)\r\n\\]\r\n\u306e\u6700\u5927\u5024 \\(m(t)\\) \u3092\u6c42\u3081\u3088. \uff08\u6700\u5927\u5024\u3092\u3068\u308b \\(x\\) \u3092\u6c42\u3081\u308b\u5fc5\u8981\u306f\u306a\u3044. \uff09<\/p><\/li>\r\n (2)<\/strong>\u3000\u3059\u3079\u3066\u306e\u5b9f\u6570 \\(t\\) \u306b\u5bfe\u3057\r\n\\[\r\n2 \\text{Tr} \\left( U(t) C U(-t) D \\right) \\geqq \\text{Tr} \\left( U(t) A U(-t) +B \\right) -m(t)\r\n\\]\r\n\u304c\u6210\u308a\u7acb\u3064\u3053\u3068\u3092\u793a\u305b.<\/p><\/li>\r\n<\/ol>\r\n (1)<\/strong><\/p>\r\n \\(P =U(t) A U(-t) -B\\) , \\(Q =U(x) \\left( \\begin{array}{cc} 1 & 0 \\\\ 0 & -1 \\end{array} \\right) U(-x)\\) \u3068\u304a\u304f\u3068\r\n\\[\\begin{align}\r\nP & = \\left( \\begin{array}{cc} \\cos t & -\\sin t \\\\ \\sin t & \\cos t \\end{array} \\right) \\left( \\begin{array}{cc} a & 0 \\\\ 0 & b \\end{array} \\right) \\left( \\begin{array}{cc} \\cos t & \\sin t \\\\ -\\sin t & \\cos t \\end{array} \\right) \\\\\r\n& \\qquad -\\left( \\begin{array}{cc} b & 0 \\\\ 0 & a \\end{array} \\right) \\\\\r\n& = \\left( \\begin{array}{cc} a \\cos t & -b \\sin t \\\\ a \\sin t & b \\cos t \\end{array} \\right) \\left( \\begin{array}{cc} \\cos t & \\sin t \\\\ -\\sin t & \\cos t \\end{array} \\right) \\\\\r\n& \\qquad -\\left( \\begin{array}{cc} b & 0 \\\\ 0 & a \\end{array} \\right) \\\\\r\n& = \\left( \\begin{array}{cc} a \\cos^2 t +b \\sin^2 t & (a-b) \\sin t \\cos t \\\\ (a-b) \\sin t \\cos t & a \\cos^2 t +b \\sin^2 t \\end{array} \\right) \\\\\r\n& \\qquad -\\left( \\begin{array}{cc} b & 0 \\\\ 0 & a \\end{array} \\right) \\\\\r\n& = (a-b) \\cos t \\left( \\begin{array}{cc} \\cos t & \\sin t \\\\ \\sin t & -\\cos t \\end{array} \\right)\r\n\\end{align}\\]\r\n\\[\\begin{align}\r\nQ & = \\left( \\begin{array}{cc} \\cos x & \\sin x \\\\ \\sin x & -\\cos x \\end{array} \\right) \\left( \\begin{array}{cc} \\cos x & \\sin x \\\\ -\\sin x & \\cos x \\end{array} \\right) \\\\\r\n& = \\left( \\begin{array}{cc} \\cos 2x & \\sin 2x \\\\ \\sin 2x & -\\cos 2x \\end{array} \\right)\r\n\\end{align}\\]\r\n\u3057\u305f\u304c\u3063\u3066\r\n\\[\\begin{align}\r\nf(x) & = \\text{Tr} (PQ) \\\\\r\n& = (a-b) \\cos t ( \\cos t \\cos 2x +\\sin t \\sin 2x \\\\\r\n& \\qquad +\\sin t \\sin 2x +\\cos t \\cos 2x ) \\\\\r\n& = 2(a-b) \\cos t \\cos ( 2x-t ) \\\\\r\n& \\leqq 2(a-b) \\left| \\cos t \\right|\r\n\\end{align}\\]\r\n\u3088\u3063\u3066\r\n\\[\r\nm(t) =\\underline{2(a-b) \\left| \\cos t \\right|}\r\n\\]\r\n (2)<\/strong><\/p>\r\n \\(R =U(t) C U(-t)\\) \u3068\u304a\u304f\u3068\r\n\\[\r\nR =\\left( \\begin{array}{cc} a^c \\cos^2 t +b^c \\sin^2 t & (a^c-b^c) \\sin t \\cos t \\\\ (a^c-b^c) \\sin t \\cos t & a^c \\cos^2 t +b^c \\sin^2 t \\end{array} \\right)\r\n\\]\r\n\u306a\u306e\u3067\r\n\\[\\begin{align}\r\n\\text{Tr} & ( U(t) C U(-t) D ) = \\text{Tr} (RD) \\\\\r\n& = b^{1-c} \\left( a^c \\cos^2 t +b^c \\sin^2 t \\right) \\\\\r\n& \\qquad +a^{1-c} \\left( a^c \\cos^2 t +b^c \\sin^2 t \\right) \\\\\r\n& = \\left( a +a^c b^{1-c} \\right) \\cos^2 t +\\left( b +a^{1-c} b^c \\right) \\sin^2 t \\\\\r\n& = a+b +\\left( -a-b +a^cb^{1-c} +a^{1-c}b^c \\right) \\cos^2 t\r\n\\end{align}\\]\r\n\u307e\u305f\r\n\\[\\begin{align}\r\n\\text{Tr} & (U(t) A U(-t) +B) \\\\\r\n& = \\left( a \\cos^2 t +b \\sin^2 t +b \\right) +\\left( a \\cos^2 t +b \\sin^2 t +a \\right) \\\\\r\n& = 2(a+b)\r\n\\end{align}\\]\r\n\u3053\u308c\u3089\u3092\u7528\u3044\u3066, \u793a\u3059\u3079\u304d\u4e0d\u7b49\u5f0f\u3092\u5909\u5f62\u3059\u308b\u3068\r\n\\[\\begin{align}\r\n& 2 \\text{Tr} \\left( U(t) C U(-t) D \\right) \\\\\r\n& \\qquad \\geqq \\text{Tr} \\left( U(t) A U(-t) +B \\right) -m(t) \\\\\r\n& 2(a+b) +2 \\left( -a-b +a^cb^{1-c} +a^{1-c}b^c \\right) \\cos^2 t \\\\\r\n& \\qquad \\geqq 2(a+b) -2(a-b) \\left| \\cos t \\right| \\\\\r\n& \\left( -a-b +a^cb^{1-c} +a^{1-c}b^c \\right) \\cos^2 t \\\\\r\n& \\qquad \\geqq -(a-b) \\left| \\cos t \\right| \\\\\r\n& \\text{\u2234} \\quad a-b \\geqq \\left( a+b -a^cb^{1-c} -a^{1-c}b^c \\right) \\left| \\cos t \\right| \\ ... [ \\text{A} ]\r\n\\end{align}\\]\r\n\u3057\u305f\u304c\u3063\u3066, [A] \u304c\u6210\u7acb\u3059\u308b\u3053\u3068\u3092\u793a\u305b\u3070\u3088\u3044.
\r\n\r\n\u3010 \u89e3 \u7b54 \u3011<\/h4>\r\n
\r\n\u3053\u3053\u3067\r\n\\[\\begin{align}\r\na-b & - \\left( a+b -a^cb^{1-c} -a^{1-c}b^c \\right) \\\\\r\n& = b \\left( a^c b^{-c} -1 \\right) +b \\left( a^{1-c} b^{c-1)} -1 \\right) \\\\\r\n& \\geqq 0 \\quad \\left( \\text{\u2235} \\ a^c b^{-c} \\geqq 1 , \\ a^{1-c} b^{c-1} \\geqq 1 \\right) \\\\\r\n\\text{\u2234} \\ a-b & \\geqq a+b -a^cb^{1-c} -a^{1-c}b^c\r\n\\end{align}\\]\r\n\u3053\u308c\u3068 \\(0 \\leqq \\left| \\cos t \\right| \\leqq 1\\) \u3067\u3042\u308b\u3053\u3068\u3092\u7528\u3044\u308b\u3068\r\n\\[\\begin{align}\r\na-b & \\geqq a+b -a^cb^{1-c} -a^{1-c}b^c \\\\\r\n& \\geqq \\left( a+b -a^cb^{1-c} -a^{1-c}b^c \\right) \\left| \\cos t \\right|\r\n\\end{align}\\]\r\n\u3088\u3063\u3066, [A] \u304c\u6210\u7acb\u3059\u308b\u3053\u3068\u304c\u793a\u3055\u308c, \u984c\u610f\u3082\u793a\u3055\u308c\u305f.<\/p>\r\n","protected":false},"excerpt":{"rendered":"\\(2 \\times 2\\) \u884c\u5217 \\(P =\\left( \\begin{array}{cc} p & q \\\\ r & s \\end{array} \\right)\\) \u306b\u5bfe\u3057\u3066 \\[ \\text{Tr}(P) =p+s … \u7d9a\u304d\u3092\u8aad\u3080