Systems with global phase invariance, 3D string?

In summary, the 3D string starts out as a solid in S^3, movement occurs in S^1, and the space of R^1 is added at each point on S^1 for "plotting" the velocity of a point. The 3D string may stretch as a function of space coordinates, increasing its potential energy.
  • #1

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.
 
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  • #2
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
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
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 said:
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
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.
 

1. What is a system with global phase invariance?

A system with global phase invariance is a type of physical or mathematical system in which the overall phase of the system does not affect its behavior. This means that any changes in the phase of the system do not alter its properties or dynamics.

2. How does global phase invariance impact 3D strings?

In the context of 3D strings, global phase invariance means that the overall phase of the string does not affect its behavior. This allows for the study and manipulation of 3D strings without the need to consider or account for changes in phase.

3. What are the applications of systems with global phase invariance and 3D strings?

Systems with global phase invariance and 3D strings have a wide range of potential applications, including in areas such as quantum mechanics, condensed matter physics, and string theory. They can also be used in engineering and technology, such as in the development of advanced materials and devices.

4. How are systems with global phase invariance and 3D strings studied?

Scientists use various mathematical and computational techniques to study systems with global phase invariance and 3D strings. This may involve simulations, experiments, and theoretical models to understand their properties and behaviors.

5. What are the current challenges in researching systems with global phase invariance and 3D strings?

One of the main challenges in studying systems with global phase invariance and 3D strings is the complexity of these systems and the difficulty in accurately modeling and manipulating them. Another challenge is the need for more advanced and precise measurement techniques to fully understand their properties and behaviors.

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