cfgauss
Oct12-06, 04:16 AM
I have been studying tensor calculus in relativity a little bit before
school starts, and I have a question that I haven't been able to answer
by myself. I know that we can rewrite Maxwell's equations using the
electromagnetic field tensor,
F^(mu nu) =
(0 E^1 E^2 E^3)
(-E^1 0 B^3 -B^2)
(-E^2 -B^3 0 B^1)
(-E^3 B^2 -B^1 0)
And we can re-write Maxwell's equations as, with d meaning curly-d, and
J is the 4-current
d_nu F^(mu nu) = 4 pi J^mu
d_lambda F_(mu nu) + d_mu F_(nu lambda) + d_nu F_(lambda mu) = 0
With which we can do all kinds of nice special relativity-stuff, and, I
presume, general relativity stuff, too (although I haven't gotten that
far in what I've been studying).
Now, I know that in quantum mechanics, we have a complex valued state
vector Psi, satisfying some differential equation (which we'll assume
is not the Schrödinger equation, but is relativistically correct,
because I've been thinking about relativity). We can write Psi in
terms of real and imaginary parts, say Psi = A + i B. Now, presumably
if we wanted we could re-write the differential equation that Psi
satisfies as two separate differential equations, one in terms of A and
one in terms of B. Now, it seems like we could form another "quantum
mechanical field tensor" in terms of the components of A and B, just
like we did for E and B, and re-write our differential equations as
tensor equations like we did before, and do more relativity-stuff.
Now, it seems to me that since we've got something in the tensor
language of general relativity, we should be able to do general
relativity with this. But, obviously, we can't, or people would be
doing this. Why don't we do this (or do we, and no one has told me
about it)? At what point does this break and not make any sense
anymore? Can we at least do relativistic quantum mechanics like this
if we wanted to?
I'd really appreciate any comments anyone has about this!
Thanks,
Jeremy Price
cfgauss@u.washington.edu
school starts, and I have a question that I haven't been able to answer
by myself. I know that we can rewrite Maxwell's equations using the
electromagnetic field tensor,
F^(mu nu) =
(0 E^1 E^2 E^3)
(-E^1 0 B^3 -B^2)
(-E^2 -B^3 0 B^1)
(-E^3 B^2 -B^1 0)
And we can re-write Maxwell's equations as, with d meaning curly-d, and
J is the 4-current
d_nu F^(mu nu) = 4 pi J^mu
d_lambda F_(mu nu) + d_mu F_(nu lambda) + d_nu F_(lambda mu) = 0
With which we can do all kinds of nice special relativity-stuff, and, I
presume, general relativity stuff, too (although I haven't gotten that
far in what I've been studying).
Now, I know that in quantum mechanics, we have a complex valued state
vector Psi, satisfying some differential equation (which we'll assume
is not the Schrödinger equation, but is relativistically correct,
because I've been thinking about relativity). We can write Psi in
terms of real and imaginary parts, say Psi = A + i B. Now, presumably
if we wanted we could re-write the differential equation that Psi
satisfies as two separate differential equations, one in terms of A and
one in terms of B. Now, it seems like we could form another "quantum
mechanical field tensor" in terms of the components of A and B, just
like we did for E and B, and re-write our differential equations as
tensor equations like we did before, and do more relativity-stuff.
Now, it seems to me that since we've got something in the tensor
language of general relativity, we should be able to do general
relativity with this. But, obviously, we can't, or people would be
doing this. Why don't we do this (or do we, and no one has told me
about it)? At what point does this break and not make any sense
anymore? Can we at least do relativistic quantum mechanics like this
if we wanted to?
I'd really appreciate any comments anyone has about this!
Thanks,
Jeremy Price
cfgauss@u.washington.edu