projective geometry in theoretical physics
rst
I have read that Dirac used projective geometry in his derivations ...
But however in all physics books I have seen there are no use of
projective geometry .. Why ?
If in any case projective geometry is used in physics could anybody
recommend a book ?
Thank you in advance ...
per.v...@gmail.com
> I have read that Dirac used projective geometry in his derivations
...
> But however in all physics books I have seen there are no use of
> projective geometry ..
Most textbooks don't explicitly identify it as such. However, I'm sure
you've run into the idea that states are identified with rays (rather
than vectors) in a complex Hilbert space. That is, a pair of state
vectors u and v in the complex Hilbert space are considered physically
equivalent if and only if there exists a nonzero complex number lambda
such that u = lambda v. But this is just another way of saying that the
true state space is the projectivization of the Hilbert space,
resulting in a complex projective space.
We can do calculations in this complex projective space by doing
calculations in the underlying Hilbert space as long as we always keep
the identification between complex-parallel vectors in mind. What
physicists usually do is work with unit vectors and then identify unit
vectors that differ by a phase change, corresponding to a factor of a
unit complex number (which can be represented as exp(it) for some real
number t, this real number being interpreted as the phase difference).
Anyway, I'm sure you see how this restriction to unit vectors is
natural, given the usual probabilistic interpretation of the vectors as
probability amplitudes.
I am not a physicist and so if any of this is a misrepresentation, I'd
love to be corrected by someone more knowledgable.
Per
antima...@yahoo.com
Well, there is direct projective geometry, in two ways (one and two
halves);
1) Quantum mechanics is projective - we don't worry about the phase of
the wave function, so the components of the state vector have a
homogeneous aspect
2a) Kaluza-Klein theory is - accidentally - based on something called
by its creators the "projective geometry of paths". See the literature
from the 30s by T Y Thomas, L P Eisenhart, O Veblen etc.
2b) From the perspective of projective geometry, the difference between
Minkowski (affine) geometry and Euclidean (affine) geometry is that the
former is characterized by a real number (c=1) and the latter by an
imaginary one (c=i). So relativity is a nearly perfect realization of
the idea of "projective metric" in the sense of Klein's program. (Not
the same Klein as 2a).
However, what never appear in physics (as far as I can tell) are
projective invariants, that is, 4-point linear fractional invariants
(cross ratios). I've always thought that this was odd. If one could
make a thoroughgoing projective physics, the problem with infinities of
all kinds would go completely away.
-drl
rst
per.v...@gmail.com wrote:
> rst wrote:
> > I have read that Dirac used projective geometry in his derivations
> ...
> > But however in all physics books I have seen there are no use of
> > projective geometry ..
>
> Most textbooks don't explicitly identify it as such.
Could you recommend a book about use projective geometry in physics ..?
Van www
per.v...@gmail.com wrote:
> rst wrote:
> > I have read that Dirac used projective geometry in his derivations
> ...
> > But however in all physics books I have seen there are no use of
> > projective geometry ..
>
> Most textbooks don't explicitly identify it as such. However, I'm sure
> you've run into the idea that states are identified with rays (rather
> than vectors) in a complex Hilbert space. That is, a pair of state
Yes, I think that's about it. Wave functions have to be normalized
to 1, I think.
I would be interested if there is more to it too.
Van
rst
Arnold Neumaier
> However, what never appear in physics (as far as I can tell) are
> projective invariants, that is, 4-point linear fractional invariants
> (cross ratios).
This is because these are invariants of the projective line (1D),
or rather its automorphism group PSL(1), and QM on the projective line
is trivial - a single qbit, Fullfledged QM needs an infinite-dimensional
projective space, ans since there is a distinguisehd metric, the invariance
group is the group of unitary transformations, not an infinite-dimensional PSL.
Arnold Neumaier
Strong_Field
news:1102641916.8...@c13g2000cwb.googlegroups.com...
>
>
> If one could
> make a thoroughgoing projective physics, the problem with infinities of
> all kinds would go completely away.
What are these infinities? How many are there? Why do they exist?
antima...@yahoo.com
> > If one could
> > make a thoroughgoing projective physics, the problem with
infinities of
> > all kinds would go completely away.
>
> What are these infinities? How many are there? Why do they exist?
Because like thermodynamics, which is inherently free of infinities,
when one deals with homogeneous functions of degree 1, there is an
implied independence from scales. I've never seen projective
thermodynamics but something like it must exist. Assuming something
like that, the formula
S = k log W
which looks exactly like a Klein-Laguerre projective measure, would
cause one to think a lot about the scaling behavior of the number of
states, that is, the density of states.
-drl
Eugene Stefanovich
I would recommend
C.Piron, "Foundations of Quantum Physics" (W. A. Benjamin, Reading, 1976)
where quantum mechanics is formulated in the language of
(quantum) logic and projective geometry.
Eugene Stefanovich.
Van www
I would say that Euclidean geometry is elliptic in that it
has a metric g = diag(1,1,1) in 3D,
while Minkowski geometry is hyperbolic with g = diag(-1,1,1,1),
and integrates time into things.
Van
antima...@yahoo.com
Arnold Neumaier wrote:
I don't really understand this point. First, I'm referring also the the
real geometry of the world, not just the ray-nature of QM. Second, in
the real geometry of the world, we have at least two direct examples of
projective ideas (Euclidean and Minkowski space and their ideal
domains). Third, we have the very interesting fact that we don't
experience dilations, although the spaces of our "experience" are -
apparently - Euclidean and Minkowskian *affine* geometry, and allow
dilations. I strongly suspect the missing projective invariants are
somehow associated with these missing dilations.
-drl
Cl.Massé
"rst" <rust...@yahoo.com> a écrit dans le message de
news:1102592282.0...@f14g2000cwb.googlegroups.com...
> I have read that Dirac used projective geometry in his derivations ...
> But however in all physics books I have seen there are no use of
> projective geometry .. Why ?
It is used mainly in the theory of the magnetic monopole, not a so
standard topic.
--
~~~~ clmasse on free dot F-country
Liberty, Equality, Profitability.
Eugene Shubert
"rst" <rust...@yahoo.com> wrote in message
news:1102592282.0...@f14g2000cwb.googlegroups.com...
> If in any case projective geometry is used in physics could anybody
> recommend a book ?
> Thank you in advance ...
Yes.
See http://homepage.ntlworld.com/stebla/Whitehead.html
and http://homepage.ntlworld.com/stebla/papers/Euclid.pdf
Eugene Shubert
http://www.everythingimportant.org
Strong_Field
news:1102992489.5...@z14g2000cwz.googlegroups.com...
Maybe I should have asked a different question. Are infinities in physics
caused by a particular choice of geometry? Would they disappear if different
geometry is used?
>
antima...@yahoo.com
> Maybe I should have asked a different question. Are infinities in
physics
> caused by a particular choice of geometry? Would they disappear if
different
> geometry is used?
Not as such, but PG allows one to consistently deal with infinity in a
continuum.
-drl
Arnold Neumaier
> Arnold Neumaier wrote:
>
>
>>>However, what never appear in physics (as far as I can tell) are
>>>projective invariants, that is, 4-point linear fractional
>
> invariants
>
>>>(cross ratios).
>>
>>This is because these are invariants of the projective line (1D),
>>or rather its automorphism group PSL(1), and QM on the projective
>
> line
>
>>is trivial - a single qbit, Fullfledged QM needs an
>
> infinite-dimensional
>
>>projective space, ans since there is a distinguisehd metric, the
>
> invariance
>
>>group is the group of unitary transformations, not an
>
> infinite-dimensional PSL.
>
> I don't really understand this point. First, I'm referring also the the
> real geometry of the world, not just the ray-nature of QM.
real geometry is 3-dimension and Euclidean, or 4-dimesnional and Minkowski.
They have their own symmetry groups - I don't see how the group PSL(1) and
cross ratios could be relevant for real 3D or 4D geometry.
Projective geomentry only enters when viewing 3D objects from 2D cameras,
and this is well understood.
> Second, in
> the real geometry of the world, we have at least two direct examples of
> projective ideas (Euclidean and Minkowski space and their ideal
> domains).
There is nothing projective in Euclidean and Minkowski space.
Projective transformations change the metric.
> Third, we have the very interesting fact that we don't
> experience dilations, although the spaces of our "experience" are -
> apparently - Euclidean and Minkowskian *affine* geometry, and allow
> dilations. I strongly suspect the missing projective invariants are
> somehow associated with these missing dilations.
Even augmenting the group of Euclidean motions by dilations leaves you
far away from the projective group!
Speculation in physics should be constrained by the requirements of
reality and the demands of compatible mathematics.
Arnold Neumaier
Arnold Neumaier
antima...@yahoo.com
> real geometry is 3-dimension and Euclidean, or 4-dimesnional and Minkowski.
> They have their own symmetry groups - I don't see how the group PSL(1) and
> cross ratios could be relevant for real 3D or 4D geometry.
Ok, give two events, they determine a line, which intersects the light
cone twice, thus 4 collinear points, so there is the cross-ratio
> Projective geomentry only enters when viewing 3D objects from 2D cameras,
> and this is well understood.
See above.
> There is nothing projective in Euclidean and Minkowski space.
> Projective transformations change the metric.
Euclid: (x - iy)(x + iy) = 0
Minkowski: (x - ct)(x + ct) = 0
> > Third, we have the very interesting fact that we don't
> > experience dilations, although the spaces of our "experience" are -
> > apparently - Euclidean and Minkowskian *affine* geometry, and allow
> > dilations. I strongly suspect the missing projective invariants are
> > somehow associated with these missing dilations.
>
> Even augmenting the group of Euclidean motions by dilations leaves you
> far away from the projective group!
This misses the point entirely. E and M are *affine*, not projective,
geometries. We don't experience (at least naively) one characteristic
feature of affine space, the dilations. So the space of experience is
*not* affine.
> Speculation in physics should be constrained by the requirements of
> reality and the demands of compatible mathematics.
Relativity = quadratic forms, any way you cut it.
-drl
Arnold Neumaier
> Arnold Neumaier wrote:
>
>>real geometry is 3-dimension and Euclidean, or 4-dimesnional and Minkowski.
>>They have their own symmetry groups - I don't see how the group PSL(1) and
>>cross ratios could be relevant for real 3D or 4D geometry.
>
> Ok, give two events, they determine a line, which intersects the light
> cone twice, thus 4 collinear points, so there is the cross-ratio
What should be its physical interpretation???
>>>Third, we have the very interesting fact that we don't
>>>experience dilations, although the spaces of our "experience" are -
>>>apparently - Euclidean and Minkowskian *affine* geometry, and allow
>>>dilations. I strongly suspect the missing projective invariants are
>>>somehow associated with these missing dilations.
>>
>>Even augmenting the group of Euclidean motions by dilations leaves you
>>far away from the projective group!
>
> This misses the point entirely. E and M are *affine*, not projective,
> geometries. We don't experience (at least naively) one characteristic
> feature of affine space, the dilations. So the space of experience is
> *not* affine.
Dilation invariance is broken by the existence of objects, which provide
a mass and length scale. Thus one would not expect to experience it.
Anyway, this has nothing to do with projective geometry.
Arnold Neumaier
antima...@yahoo.com
> > Ok, give two events, they determine a line, which intersects the light
> > cone twice, thus 4 collinear points, so there is the cross-ratio
>
> What should be its physical interpretation???
This one is simple - log XR ~ proper time between the events. This is
the standard embedding of a (pseudo) Pythagorean metric inside
projective geometry, using a given quadratic form (in this case, a
degenerate one = affine space).
Likewise - in Euclidean geometry the "light cone" is better known as
the "circular points at infinity" with equation
x^2 + y^2 + z^2 + ... = 0
Give two points, their connecting line intersects this "ideal domain"
in two points, form XR, take log, multiply by i, result is the
Pythagorean distance between the points.
Somehow the observer posits the ideal domain - in one case, the light
cone, in the other (with c->inf), "spatial infinity", that is, where
parallel rails meet! The "thoroughgoing projective physics" would be
invariant under changing this domain.
-drl
antima...@yahoo.com
Van www wrote:
> I would say that Euclidean geometry is elliptic in that it
> has a metric g = diag(1,1,1) in 3D,
> while Minkowski geometry is hyperbolic with g = diag(-1,1,1,1),
> and integrates time into things.
Actually both are affine, because the fundamental quadric is
degenerate, e.g. in the plane
Minkowski (x-ct)(x+ct) = 0 (light cone)
Euclid (x-iy)(x+iy) = 0 (circular points at infinity)
BTW for the person asking about books, the classic is
"Nicht-Euklidische Geometrie" by Felix Klein, which however is in
German (until I get around to translating it :)
-drl