Excel-VBAによる SU(3)

Excel-VBA による SU(3) の構造定数の計算
ゲルマン行列 λa , SU(3) の構造定数 fabc に対して、
tr ([λa , λb ] λc ) = 4ifabc
でした。これを利用して fabc を Excel-VBA で数値計算します。以下はそのソー
スです。(アポストロフィ ’ に続く文は REM 文です。Dim で宣言されていない
変数は Variant 型で、型の選択をインタプリタに委ねます。これにより BASIC 言
語らしく宣言文を少なくできるわけです。)
Function gellmann(a) As matrix ’ ゲルマン行列
Dim g As matrix
g = zeromatrix(3) ’g を 3 次零行列とする
If a = 1 Then g.ele(1, 2) = cn(1, 0): g.ele(2, 1) = cn(1, 0) ’ 行列要素修正
If a = 2 Then g.ele(1, 2) = cn(0, -1): g.ele(2, 1) = cn(0, 1)
If a = 3 Then g.ele(1, 1) = cn(1, 0): g.ele(2, 2) = cn(-1, 0)
If a = 4 Then g.ele(1, 3) = cn(1, 0): g.ele(3, 1) = cn(1, 0)
If a = 5 Then g.ele(1, 3) = cn(0, -1): g.ele(3, 1) = cn(0, 1)
If a = 6 Then g.ele(2, 3) = cn(1, 0): g.ele(3, 2) = cn(1, 0)
If a = 7 Then g.ele(2, 3) = cn(0, -1): g.ele(3, 2) = cn(0, 1)
If a = 8 Then
g.ele(1, 1) = cn(1, 0): g.ele(2, 2) = cn(1, 0): g.ele(3, 3) = cn(-2, 0)
g = cmproduct(cn(3 ^ (-0.5), 0), g) ’g → (1/√ 3)g
End If
gellmann = g
End Function
Function tstr3(a, b, c) As String ’3 数値の連続文字化
tstr3 = Trim(Str(a)) + Trim(Str(b)) + Trim(Str(c))
End Function
Function ctostr(a As complex) As String ’ 複素数の定型文字化
ctostr = "(" + Format(a.re, "0.000000") + ", " + Format(a.im, "0.000000") + ")"
End Function
Sub main()
Dim λ (8) As matrix, f As complex
For a = 1 To 8
λ (a) = gellmann(a)
Next a
i = 1
For a = 1 To 8: For b = a + 1 To 8: For c = b + 1 To 8
f = cproduct(cn(0, -0.25), trace(mproduct(commutator(λ (a), λ (b)), λ (c))))
’ ↑ f = (-i/4) tr( [λ (a), λ (b)] λ (c) )
If Not (f.re = 0 And f.im = 0) Then ’f ≠ 0 のときだけ出力
i = i + 1
Cells(i, 2).Value = "f[" + tstr3(a, b, c) + "]="
Cells(i, 3).Value = ctostr(f) ’ 複素数 f を文字化したものを出力
End If
Next c: Next b: Next a
i = i + 2
Cells(i, 2).Value = "√ 3/2=": Cells(i, 3).Value = 3 ^ 0.5 / 2 ’ 参照値出力
End Sub
main 実行で Excel のシート上に以下の出力が得られます。
図 1: 出力
複素数や複素行列の型 (構造体) およびこれらに関連した関数は以下のソースで
与えられています (標準モジュール用)。
Type complex ’ 複素数の型
re As Double ’ 実部
im As Double ’ 虚部
End Type
Type matrix ’ 複素行列の型
dim As Integer ’ 次元
ele(10, 10) As complex ’ 要素
End Type
Function cn(a As Double, b As Double) As complex ’ 複素数の実表示
cn.re = a: cn.im = b
End Function
Function cplus(a As complex, b As complex) As complex ’ 複素数の和
cplus.re = a.re + b.re: cplus.im = a.im + b.im
End Function
Function cminus(a As complex, b As complex) As complex ’ 複素数の差
cminus.re = a.re - b.re: cminus.im = a.im - b.im
End Function
Function cproduct(a As complex, b As complex) As complex ’ 複素数の積
cproduct.re = a.re * b.re - a.im * b.im: cproduct.im = a.re * b.im + a.im * b.re
End Function
Function zeromatrix(d) As matrix ’ 零行列
zeromatrix.dim = d
For i = 1 To d: For j = 1 To d
zeromatrix.ele(i, j) = cn(0, 0)
Next j: Next i
End Function
Function mplus(a As matrix, b As matrix) As matrix ’ 行列の和
mplus.dim = a.dim
For i = 1 To a.dim: For j = 1 To a.dim
mplus.ele(i, j) = cplus(a.ele(i, j), b.ele(i, j))
Next j: Next i
End Function
Function mminus(a As matrix, b As matrix) As matrix ’ 行列の差
mminus.dim = a.dim
For i = 1 To a.dim: For j = 1 To a.dim
mminus.ele(i, j) = cminus(a.ele(i, j), b.ele(i, j))
Next j: Next i
End Function
Function cmproduct(c As complex, a As matrix) As matrix ’ 複素数と行列の積
cmproduct.dim = a.dim
For i = 1 To a.dim: For j = 1 To a.dim
cmproduct.ele(i, j) = cproduct(c, a.ele(i, j))
Next j: Next i
End Function
Function mproduct(a As matrix, b As matrix) As matrix ’ 行列の積
Dim s As complex
mproduct.dim = a.dim
For i = 1 To a.dim: For j = 1 To a.dim
s = cn(0, 0)
For k = 1 To a.dim
s = cplus(s, cproduct(a.ele(i, k), b.ele(k, j)))
Next k
mproduct.ele(i, j) = s
Next j: Next i
End Function
Function trace(a As matrix) As complex ’ トレース
Dim s As complex
s = cn(0, 0)
For i = 1 To a.dim
s = cplus(s, a.ele(i, i))
Next i
trace = s
End Function
Function commutator(a As matrix, b As matrix) As matrix ’ 交換子
commutator = mminus(mproduct(a, b), mproduct(b, a))
End Function
以上。

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