Real or Complex space ?
pre...@my-deja.com
is best modeled ( in GR ) by a real 4-dimensional
manifold rather than a complex 2-dimensional one.
Or are these mathematically the same ( *morphic ?)
Thanks.
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torqu...@my-deja.com
pre...@my-deja.com wrote:
> How do we know ( experimentally ) that space-time
> is best modeled ( in GR ) by a real 4-dimensional
> manifold rather than a complex 2-dimensional one.
>
> Or are these mathematically the same ( *morphic ?)
A complex manifold comes with a structure that defines what it means to
say that a function on it is holomorphic. This is part of what makes a
manifold interesting - the set of (locally) holomorphic functions on it.
On the other hand a real manifold comes with the set of continuous (or
differentiable if you really mean 'differentiable manifold') functions.
What we generally observe around us are functions that are continuous
(or differentiable) but not holomorphic. Also the complex structure of a
2 dimensional complex manifold isn't really compatible with the SO(1,3)
(Lorentz) symmetry that we observe in the universe.
--
Torque
http://travel.to/tanelorn
ba...@galaxy.ucr.edu
>SL(2,C) is the covering group for SO(3,1) and they have the same local
>structure. I was thinking maybe the existence of spinor fields suggested
>we lived in a complex rather than real space.
Well, SL(2,C) acts on the space of spinors (namely C^2), while
SO(3,1) acts on Minkowski spacetime (namely R^4). Since both C^2
and R^4 are 4-dimensional as *real* vector spaces, you might hope
that the space of spinors and Minkowski spacetime are the same thing
in disguise. Maybe this is why you were tempted to treat spacetime
as a 2-dimensional complex manifold. But Minkowski spacetime and
the space of spinors are very different. First of all, SO(3,1)
acts on the former but not the latter. Second of all, SO(3,1)
preserves a *symmetric real* bilinear form on Minkowski spacetime,
while SL(2,C) preserves a *skew-symmetric complex* bilinear form on
the space of spinors. So they are very different - though deeply
related - beasts.
You might have more luck if you complexified Minkowski spacetime.
Check out the work of Penrose.
pre...@my-deja.com
ba...@galaxy.ucr.edu wrote:
> Well, SL(2,C) acts on the space of spinors (namely C^2), while
> SO(3,1) acts on Minkowski spacetime (namely R^4). Since both C^2
> and R^4 are 4-dimensional as *real* vector spaces, you might hope
> that the space of spinors and Minkowski spacetime are the same thing
> in disguise. Maybe this is why you were tempted to treat spacetime
> as a 2-dimensional complex manifold. But Minkowski spacetime and
> the space of spinors are very different. First of all, SO(3,1)
> acts on the former but not the latter. Second of all, SO(3,1)
> preserves a *symmetric real* bilinear form on Minkowski spacetime,
> while SL(2,C) preserves a *skew-symmetric complex* bilinear form on
> the space of spinors. So they are very different - though deeply
> related - beasts.
>
> You might have more luck if you complexified Minkowski spacetime.
> Check out the work of Penrose.
>
>
which one I live in? Experimentally I know I'm in a *real* 4-dimensional
vector space ( because I can look at boundaries of points in boundaries
.... etc ). But if I look at Lorentz transformations then for each
Lorentz transformation a can pick an element of SO(3,1) or equally an
element of SL(2,C) since there ia a 2->1 map between the two.
I don't know what it would be like to live in C^2 but would it not be
similar ( or identical ) to the world I see around me ?
John Baez
>Ok, so C^2 is similar (but different) from R^4, but how can I tell
>which one I live in? Experimentally I know I'm in a *real* 4-dimensional
>vector space ( because I can look at boundaries of points in boundaries
>.... etc ). But if I look at Lorentz transformations then for each
>Lorentz transformation a can pick an element of SO(3,1) or equally an
>element of SL(2,C) since there is a 2->1 map between the two.
What I was trying to tell you in my previous post was that the spacetime
we live in is obviously R^4 with its usual action of SO(3,1), not C^2
with its usual action of SL(2,C), because they really are completely
different. There is no way we could mistake the two. I explained
a bunch of reasons why. Basically, mistaking R^4 for C^2 amounts to
mistaking vectors for spinors. They transform in completely different
ways under Lorentz transformations.
>I don't know what it would be like to live in C^2 but would it not be
>similar ( or identical ) to the world I see around me ?
No, it would be very different. To see yet another reason: if
spacetime were C^2, then for each direction v there would be a
special direction called "i times v". There's not.
Jackie
> SL(2,C) is the covering group for SO(3,1) and they have the same local
> structure. I was thinking maybe the existence of spinor fields suggested
> we lived in a complex rather than real space.
You may be limiting yourself by thinking of 2 dimensions.
I would think that S0(3,1) is a subspace of C^4.
And if we go back to physics and take c as a maximum velocity, then
particles travelling at relative velocities greater than c would not be
able to exchange signals. At any given time, any observer could only
ever observe a(n ever changing) SO(3,1) slice of this C^4 space.
In such a Universe, observation would tend to confirm that the Universe
was best described by SO(3,1). The C^4 embedding space might reveal
itself most clearly at small scales and high energy levels.
And, of course, a C^4 Universe could have a rotation, in which every
direction we looked would be in the direction of the "equator".
Barry
John Baez
> Is there any way to generalize Fourier
>Transforms to functions on spaces other than R^n?
>If so, which spaces work? That is, to what extent
>can you define momentum/energy in the context of
>Quantum Mechanics on curved spaces/spacetimes?
A very good question. People have been working for the
last century or so to answer this question; the stuff
they came up with is called "harmonic analysis".
Some of the nice stuff about Fourier transforms works
because R^n is a locally compact abelian group - this
stuff is called "Poyntragin duality". A good example
is the fact that the Fourier transform of a function
on the circle is a function on the integers, and conversely,
the Fourier transform of a function on the integers is a
function on the circle. We say that the circle is the
"Pontryagin dual" of the integers and vice versa. R^n is
the Pontryagin dual of itself.
Some of the nice stuff works just because R^n is a Lie group
- this stuff is called harmonic analysis on Lie groups. A
good example is how you can decompose any function on a
compact Lie group like SU(n) or SO(n) into a linear
combination of matrix elements of irreducible representations.
This is called the Peter-Weyl theorem. Noncompact nonabelian
groups like the Lorentz groups SO(n,1) are a bit harder,
but a lot has been done with them too, especially starting
with the work of Harish-Chandra (a student of Dirac who
started as a physicist but switched to math).
Some of the nice stuff works simply because R^n is a
homogeneous space - this stuff is called harmonic analysis
on homogeneous spaces. A homogeneous space is a space of
the form G/H where G is a Lie group and H is a subgroup.
A good example is the sphere S^2 = SO(3)/SO(2). We can
write any function on the sphere as a linear combination
of spherical harmonics. This kind of thing has been
generalized to all sorts of homogeneous spaces.
And some of the nice stuff works just because R^n is a
manifold - this stuff is called the theory of
pseudodifferential operators and Fourier integral operators.
The idea here is that even if your manifold doesn't have much
symmetry at all, it still looks *locally* like R^n, so you can
do a kind of local analogue of Fourier analysis on it.
Basically, the more symmetry your space has, the easier
it is to do something like Fourier analysis on it. Above
I listed 4 of the main branches of harmonic analysis, in
order of decreasing symmetry.
There is a lot to read on this subject - almost too much,
in fact! - and any mathematical physicist worth their salt
winds up spending years studying this material. However,
here's a great place to start:
George W. Mackey, The scope and history of commutative and
noncommutative harmonic analysis, Providence, R.I., American
Mathematical Society, 1992. Series title: History of
Mathematics; v. 5.
George Mackey is the "grand old man" of harmonic analysis, so
he's a pretty good tour guide - though there's lots of cool
stuff he doesn't say much about, too.
While I'm urging Mackey's book on you, I should add that he's
written a bunch of other books on similar subjects which are
also good. Here are two:
George Mackey, Quantum mechanics from the point of view of the
theory of group representations, Mathematical Sciences Research
Institute, 1984.
George Mackey, Unitary group representations in physics, probability,
and number theory, Addison-Wesley, 1989.
The theory of group representations is closely related to harmonic
analysis, especially the way Mackey does it, so if you can't find
that other book, try these!
David M. Palmer
> Is there any way to generalize Fourier
> Transforms to functions on spaces other than R^n?
> If so, which spaces work? That is, to what extent
> can you define momentum/energy in the context of
> Quantum Mechanics on curved spaces/spacetimes?
Spherical harmonics are one such system of orthogonal basis functions
on a curved space. Probably not as general as you are thinking of.
--
David Palmer dmpa...@clark.net
http://www.clark.net/pub/dmpalmer/