mecanismo de gauge SU(3) x SU(4)

g_{x}(t) = e^{(x+(-y))·i·t}·f_{x}(t)
g_{y}(t) = e^{(y+(-z))·i·t}·f_{y}(t)
g_{z}(t) = e^{(z+(-x))·i·t}·f_{z}(t)


g_{a}(t) = e^{(a+(-b))·i·t}·f_{a}(t)
g_{b}(t) = e^{(b+(-u))·i·t}·f_{b}(t)
g_{u}(t) = e^{(u+(-v))·i·t}·f_{u}(t)
g_{v}(t) = e^{(v+(-a))·i·t}·f_{v}(t)


g_{x}(t)·g_{y}(t)·g_{z}(t)·g_{a}(t)·g_{b}(t)·g_{u}(t)·g_{v}(t) = ...
... f_{x}(t)·f_{y}(t)·f_{z}(t)·f_{a}(t)·f_{b}(t)·f_{u}(t)·f_{v}(t)


d_{t}[g_{x}(t)]·d_{t}[g_{y}(t)]·d_{t}[g_{z}(t)]·d_{t}[g_{a}(t)]·d_{t}[g_{b}(t)]·d_{t}[g_{u}(t)]·d_{t}[g_{v}(t)] = ...
... d_{t}[f_{x}(t)]·d_{t}[f_{y}(t)]·d_{t}[f_{z}(t)]·d_{t}[f_{a}(t)]·d_{t}[f_{b}(t)]·d_{t}[f_{u}(t)]·d_{t}[f_{v}(t)]+...
... (x+(-y))(y+(-z))(z+(-x))(a+(-b))(b+(-u))(u+(-v))(v+(-a))·(-i)·...
... f_{x}(t)·f_{y}(t)·f_{z}(t)·f_{a}(t)·f_{b}(t)·f_{u}(t)·f_{v}(t) = 0


f_{x}(t) = (1+x+(-y))·e^{(x+(-y))·(-i)·t}
f_{y}(t) = (1+y+(-z))·e^{(y+(-z))·(-i)·t}
f_{z}(t) = (1+z+(-x))·e^{(z+(-x))·(-i)·t}


f_{a}(t) = (1+u+(-v))·e^{(a+(-b))·(-i)·t}
f_{b}(t) = (1+v+(-a))·e^{(b+(-u))·(-i)·t}
f_{u}(t) = (1+a+(-b))·e^{(u+(-v))·(-i)·t}
f_{v}(t) = (1+b+(-u))·e^{(v+(-a))·(-i)·t}


g_{x}(t)·g_{y}(t)·g_{z}(t)·g_{a}(t)·g_{b}(t)·g_{u}(t)·g_{v}(t) = ...
... x·( y^{2}+(-1)·z^{2} )+y·( z^{2}+(-1)·x^{2} )+z·( x^{2}+(-1)·y^{2} ) +...
... (-1)·( a^{2}+b^{2}+u^{2}+v^{2} )·...
... ( x·( y^{2}+(-1)·z^{2} )+y·( z^{2}+(-1)·x^{2} )+z·( x^{2}+(-1)·y^{2} ) ) + ...
... ( a^{2}·u^{2}+b^{2}·v^{2} )·...
... ( x·( y^{2}+(-1)·z^{2} )+y·( z^{2}+(-1)·x^{2} )+z·( x^{2}+(-1)·y^{2} ) ) + ...
... ( 2·( a·b·u·v ) )·...
... ( x·( y^{2}+(-1)·z^{2} )+y·( z^{2}+(-1)·x^{2} )+z·( x^{2}+(-1)·y^{2} ) )+...
... (-1)·( a·v·( b^{2}+u^{2} )+b·u·( a^{2}+v^{2} ) )·...
... ( x·( y^{2}+(-1)·z^{2} )+y·( z^{2}+(-1)·x^{2} )+z·( x^{2}+(-1)·y^{2} ) ) + ...
... (-1)·( a·b·( u^{2}+v^{2} ) )·...
... ( x·( y^{2}+(-1)·z^{2} )+y·( z^{2}+(-1)·x^{2} )+z·( x^{2}+(-1)·y^{2} ) )

mecanismo de gauge: SU(3) x SU(2)

g_{x}(t) = e^{(x+(-y))·i·t}·f_{x}(t)
g_{y}(t) = e^{(y+(-z))·i·t}·f_{y}(t)
g_{z}(t) = e^{(z+(-x))·i·t}·f_{z}(t)


g_{u}(t) = e^{(u+(-v))·i·t}·f_{u}(t)
g_{v}(t) = e^{(v+(-u))·i·t}·f_{v}(t)


g_{x}(t)·g_{y}(t)·g_{z}(t)·g_{u}(t)·g_{v}(t) = f_{x}(t)·f_{y}(t)·f_{z}(t)·f_{u}(t)·f_{v}(t)


d_{t}[g_{x}(t)]·d_{t}[g_{y}(t)]·d_{t}[g_{z}(t)]·d_{t}[g_{u}(t)]·d_{t}[g_{v}(t)] = ...
... d_{t}[f_{x}(t)]·d_{t}[f_{y}(t)]·d_{t}[f_{z}(t)]·d_{t}[f_{u}(t)]·d_{t}[f_{v}(t)]+...
... (x+(-y))(y+(-z))(z+(-x))(u+(-v))(v+(-u))·i·f_{x}(t)·f_{y}(t)·f_{z}(t)·f_{u}(t)·f_{v}(t) = 0


d_{t}[g_{x}(t)]·d_{t}[g_{y}(t)]·d_{t}[g_{z}(t)]·d_{t}[g_{u}(t)]·d_{t}[g_{v}(t)] = 0


g_{x}(t)·g_{y}(t)·g_{z}(t)·g_{u}(t)·g_{v}(t) = ...
... x·( y^{2}+(-1)·z^{2} )+y·( z^{2}+(-1)·x^{2} )+z·( x^{2}+(-1)·y^{2} ) +...
... (-1)( u^{2}+v^{2} )·( x·( y^{2}+(-1)·z^{2} )+y·( z^{2}+(-1)·x^{2} )+z·( x^{2}+(-1)·y^{2} ) )


f_{x}(t) = (1+x+(-y))·e^{(x+(-y))·(-i)·t}
f_{y}(t) = (1+y+(-z))·e^{(y+(-z))·(-i)·t}
f_{z}(t) = (1+z+(-x))·e^{(z+(-x))·(-i)·t}


f_{u}(t) = (1+u+(-v))·e^{(u+(-v))·(-i)·t}
f_{v}(t) = (1+v+(-u))·e^{(v+(-u))·(-i)·t}


x^{2}+y^{2}+z^{2}+u^{2}+v^{2} = 1


u = (2/3) = (3/6)+(1/6)
d = (-1)(1/3) = (-1)·(3/6)+(1/6)
(-u) = (-1)(2/3) = (-1)(3/6)+(-1)(1/6)
(-d) = (1/3) = (3/6)+(-1)(1/6)

índex de física

índex de etiquetes de física

mecanisme de gauge: SU(3) cromodinámica de quarks

g_{x}(t) = e^{(x+(-y))·i·t}·f_{x}(t)
g_{y}(t) = e^{(y+(-z))·i·t}·f_{y}(t)
g_{z}(t) = e^{(z+(-x))·i·t}·f_{z}(t)


g_{x}(t)·g_{y}(t)·g_{z}(t) = f_{x}(t)·f_{y}(t)·f_{z}(t)


d_{t}[g_{x}(t)]·d_{t}[g_{y}(t)]·d_{t}[g_{z}(t)] = ...
... d_{t}[f_{x}(t)]·d_{t}[f_{y}(t)]·d_{t}[f_{z}(t)]+...
... (x+(-y))(y+(-z))(z+(-x))·(-i)·f_{x}(t)·f_{y}(t)·f_{z}(t) = 0


d_{t}[g_{x}(t)]·d_{t}[g_{y}(t)]·d_{t}[g_{z}(t)] = 0


g_{x}(t)·g_{y}(t)·g_{z}(t) = x·( y^{2}+(-1)·z^{2} )+y·( z^{2}+(-1)·x^{2} )+z·( x^{2}+(-1)·y^{2} )


f_{x}(t)·f_{y}(t)·f_{z}(t) = x·( y^{2}+(-1)·z^{2} )+y·( z^{2}+(-1)·x^{2} )+z·( x^{2}+(-1)·y^{2} )


f_{x}(t) = (1+x+(-y))·e^{(x+(-y))·(-i)·t}
f_{y}(t) = (1+y+(-z))·e^{(y+(-z))·(-i)·t}
f_{z}(t) = (1+z+(-x))·e^{(z+(-x))·(-i)·t}


x^{2}+y^{2}+z^{2} = 1


unificació:
( cos( Z·(1+(-1)q^{2}) )sin( qW ) )^{2}+( sin( Z·(1+(-1)q^{2}) )sin( qW ) )^{2}+( cos( qW ) )^{2} = 1
( cos( Z·(1+(-1)q^{2}) )cos( qW ) )^{2}+( sin( Z·(1+(-1)q^{2}) )cos( qW ) )^{2}+( sin( qW ) )^{2} = 1

mecanisme electrodébil

F(t) = e^{( qW+Z·(1+(-1)q^{2}) )·i·t}·f(t)
G(t) = e^{(-1)·( qW+Z·(1+(-1)q^{2}) )·i·t}·g(t)


F(t)·G(t) = f(t)·g(t)


d_{t}[F(t)]·d_{t}[G(t)] = d_{t}[f(t)]·d_{t}[g(t)]+( qW+Z·(1+(-1)q^{2}) )^{2}·f(t)·g(t) = 0


f(t) = e^{( qW+Z·(1+(-1)q^{2}) )·t}
g(t) = e^{(-1)·( qW+Z·(1+(-1)q^{2}) )·t}

mecanisme de gauge

f(x) = e^{qi}·P(x)
g(x) = e^{(-1)qi}·Q(x)


f(x)g(x) = P(x)Q(x)


qi·( e^{qi}·P(x)·e^{(-1)qi}·d_{x}[Q(x)] + (-1)e^{qi}·d_{x}[P(x)]·e^{(-1)qi}·Q(x) ) = 0
P(x)·d_{x}[Q(x)] + (-1)d_{x}[P(x)]·Q(x) = 0




d_{x}[f(x)]d_{x}[g(x)] = d_{x}[P(x)]d_{x}[Q(x)]+q^{2}P(x)Q(x)


d_{x}[f(x)]d_{x}[g(x)]+f(x)g(x) = d_{x}[P(x)]d_{x}[Q(x)]+(1+q^{2})P(x)Q(x)= 0


P(x) = e^{(1+q^{2})^{(1/2)}ix}
Q(x) = e^{(1+q^{2})^{(1/2)}ix}


d_{x}[f(x)]d_{x}[g(x)]+f(x)g(x) = d_{x}[P(x)]d_{x}[Q(x)]+(1+q^{2})P(x)Q(x) = (-1)qi


P(x) = cos( (1+q^{2})^{(1/2)}ix )
Q(x) = sin( (1+q^{2})^{(1/2)}ix )


P(x) = cos( (-1)(1+q^{2})^{(1/2)}ix )
Q(x) = sin( (-1)(1+q^{2})^{(1/2)}ix )


d_{x}[f(x)]d_{x}[g(x)]+f(x)g(x) = d_{x}[P(x)]d_{x}[Q(x)]+(1+q^{2})P(x)Q(x) = qi


P(x) = cosh( (1+q^{2})^{(1/2)}ix )
Q(x) = sinh( (1+q^{2})^{(1/2)}ix )




P(x) = cosh( (-1)(1+q^{2})^{(1/2)}ix )
Q(x) = sinh( (-1)(1+q^{2})^{(1/2)}ix )