Science > Physics > Energy In The Knowledge Equation 3: The Brazilian Fermion Equations
| Topic: |
Science > Physics |
| User: |
"OsherD" |
| Date: |
25 Nov 2005 12:33:32 AM |
| Object: |
Energy In The Knowledge Equation 3: The Brazilian Fermion Equations |
From Osher Doctorow
Recall that the Hubble Parameter H is:
1) H = (da(t)/dt)/a(t)
and is interpreted as the normalized rate of expansion of the Universe
(that is to say, the relative rate of expansion relative to the amount
of expansion).
The equations that Ribas et al (2005) use for what I call their
"fermion equations" are in my terminology:
2) d(rho_f)/dt + 3H(rho_f + p_f) = 0 (p. 3)
3) d(rho_m)/dt + 3H(rho_m + p_m) = -3Ho-bar
4) tau o-bar + o-bar = -3 eta H
5) the Dirac equation in the form (p. 4): dw/dt + (3/2)Hw + imw*
-2i(w*.w)w*V' - 2i(w#.w)w#V* = 0
Here m is matter or matter field, f is fermion or fermionic field.
I'll define the other quantities shortly (hopefully).
Osher Doctorow
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| User: "OsherD" |
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| Title: Re: Energy In The Knowledge Equation 3: The Brazilian Fermion Equations |
25 Nov 2005 12:46:23 AM |
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From Osher Doctorow
tau is a characteristic time, o-bar is non-equilibrium pressure, eta is
"bulk viscosity coefficient", w is a 4-component spinor w = (w1, w2,
w3, w4) or its transpose, w* has -w3 and -w4 replacing w3, w4
respectively, and w# is (w3, w4, -w1, -w2) in place of (w1, w2, w3, w4)
(or their respective transposes).
The dot product is indicated by w*.w and w#.w. V is the potential
density of self-interaction between the fermions. V' is the partial
derivative of V with respect to (w-bar w)^2 where w-bar is the complex
conjugate of w. V# is the partial derivative of V with respect to
(w-bar g^5 w)^2 with g^5 the Dirac gamma matrix.
I left out the acceleration equation, which I'll try to add next time.
Osher Doctorow
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| User: "OsherD" |
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| Title: Re: Energy In The Knowledge Equation 3: The Brazilian Fermion Equations |
25 Nov 2005 12:52:06 AM |
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From Osher Doctorow
The acceleration equation is:
1) Dtt(a)/a = -(1/6)(rho_f + rho_m + 3p_f + 3p_m + 3o-bar)
The equations in the order listed beyond the Hubble parameter are the
evolution equation of the energy density of the fermionic field, of the
matter field, of the non-equilibrium pressure, the acceleration
equation, and the Dirac equation in terms of the spinor w.
Osher Doctorow
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