Systems with global phase invariance, 3D string?

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Spinnor
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From:

http://en.wikipedia.org/wiki/Quantum_mechanics#Mathematical_formulation

...
In the mathematically rigorous formulation of quantum mechanics, developed by Paul Dirac[7] and John von Neumann[8], the possible states of a quantum mechanical system are represented by unit vectors (called "state vectors") residing in a complex separable Hilbert space (variously called the "state space" or the "associated Hilbert space" of the system) well defined up to a complex number of norm 1 (the phase factor).
...

Let a three dimensional analog of a one dimensional string completely occupy the space S^3. Movement only occurs in some tangential space, let this space be S^1. This system has a type of global phase invariance, an equal global change of the coordinate in the space S^1. Any point P in S^3 labels a point of our "3D string". An additional coordinate gives the position of point P in our space S^1. If all points in S^3 have the same coordinate in S^1 then the system will be at rest and will be in a minimum energy configuration. The state of this system does not change with a global coordinate transformation.

Now give the 3D string some energy and then freeze time. The physics is contained in the 3 dimensional surface that occupies the space S^3 U S^1, the union of S^3 and S^1. Again a global phase change does not change the physics.

Thanks for any thoughts.
 

Answers and Replies

  • #2
Spinnor
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andy everett wrote:

> From:
>
> http://en.wikipedia.org/wiki/Quantum_mechanics#Mathematical_formulation
>
>
> ...
> In the mathematically rigorous formulation of quantum mechanics,
> developed by Paul Dirac[7] and John von Neumann[8], the possible states
> of a quantum mechanical system are represented by unit vectors (called
> "state vectors") residing in a complex separable Hilbert space
> (variously called the "state space" or the "associated Hilbert space" of
> the system) well defined up to a complex number of norm 1 (the phase
> factor).
> ...
>
> Let a three dimensional analog of a one dimensional string completely
> occupy the space S^3. Movement only occurs in some tangential space, let
> this space be S^1. This system has a type of global phase invariance, an
> equal global change of the coordinate in the space S^1. Any point P in
> S^3 labels a point of our "3D string". An additional coordinate gives
> the position of point P in our space S^1.

We need another "coordinate", time.


> If all points in S^3 have the
> same coordinate in S^1

for all time

> then the system will be at rest and will be in a
> minimum energy configuration. The state of this system does not change
> with a global coordinate transformation

of the coordinate in S^1.

.
>
> Now give the 3D string some energy and then freeze time.

We imagine the 3D string will evolve with time. For each point on S^1 we must have a tangential space (R^1 works) to plot the velocity of our point P in our space S^1.


> The physics is
> contained in the 3 dimensional surface that occupies the space S^3 U
> S^1, the union of S^3 and S^1.

Not right.


The physics is contained in 4 dimensional surface in the space:

S^3 U S^1 U R^1 the union of the spaces S^3, S^1, and R^1.

> Again a global phase change

of the coordinate of S^1

> does not
> change the physics.
>
> Thanks for any thoughts.

I hope I got it right this time, thanks for any thoughts.
 
  • #3
Spinnor
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andy everett wrote:
> andy everett wrote:
>
...
>
> > The physics is contained in the 3 dimensional surface that occupies
> the space S3 U S1, the union of S3 and S1.
>
>
> Not right.
>
>
> The physics is contained in 4 dimensional surface in the space:
>
> S3 U S1 U R1 the union of the spaces S3, S1, and R1. ...

Still not right.


Our 3D string starts out as a 3D solid in S^3. Movement occurs in the space S^1. We add the space of the real number line, R^1, at each point on S^1 so we can "plot" the velocity of a point in S^1. The 3D string moves in the space S^3 U S^1 U R^1. Just as a 2 dimensional sheet of paper stays 2 dimensional as we move it about in 3 dimensions our 3D string also stays 3 dimensional as it moves in S^3 U S^1 U R^1. It may and will stretch. So the above quote should read:

"The physics is contained in the 3 dimensional surface that occupies the space S^3 U S^1 R^1, the union of S^3, S^1, and R^1."

The surface may get "twisted" as a function of space coordinates S^3, increasing potential energy of the 3D string. Each small volume of the 3D string may have some velocity in the space S^1 resulting in kinetic energy of the 3D string. So we have a simple dynamic system where physics does not change under a global rotation of a coordinate. I'm sure there are many other such realizations.

Thank you for any thoughts.
 
  • #4
Spinnor
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I wrote:

"... So we have a simple dynamic system where physics does not change under a global rotation of a coordinate. ..."

Because of this do we get some conserved quantity?

Thanks for any thoughts.
 
  • #5
Spinnor
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I wrote:

"... So we have a simple dynamic system where physics does not change under a global rotation of a coordinate. ..."

Because of this do we get some conserved quantity?

Thanks for any thoughts.
Would that be conservation of momentum in the space S^1?
 
  • #6
Spinnor
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Can we produce dipole like radiation with our 3D string? I think so.

Grab two points, P1 and P2, of our 3D string that are "near" each other. Move each point such that their coordinates in S^1 vary with time as:

P1 = d*sin(w*t)
P2 = -d*sin(w*t)

where d is much smaller than the radius r of S^1. For points of our 3D string an equal distance from P1 and P2 the forces will cancel and for those points the coordinate of S^1 will not change with time. But for all other points the forces will not cancel and we will have outward going waves.

Thanks for any thoughts.
 

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