Can Quantum Mechanics be seen as pure geometry ?

[nameta9]

Theoretical Physics in the end is simply a set of numbers (mass, impulse, energy etc.) related to another set through a series of mathematical operations. Has it ever been tried to just set all the physical quantities of elementary particles in an n-dimensional space where each dimension is the axis of a given quantity and just follow the path of the point and points in this space as they interact ? For example a particle may have n quantities (x,y,z,t,mass,spin,energy,impulse,size of particle, etc.) so you set each dimension of the space to correspond to each quantity, so aside from the typical 4 dimensions of space-time you would have other dimensions corresponding to the energy, another to the mass, another to the velocity, another to spin etc. You would end up with an n-dimensional space where the particle would simply be the path of the point in this space. Could this simplify theoretical physics ? Is this result known ?

The idea of putting every physical quantity an elementary particle can have on a different axis of an n-dimensional coordinate system may really allow some new research and insight.

 

[Berislav]

You were already beaten to the idea:

Mathematical formulation of quantum physics

Not exactly as you imagined it, but there are similarities.

 

[ZapperZ]

You were already beaten to the idea:

Mathematical formulation of quantum physics

Not exactly as you imagined it, but there are similarities.

Actually, it's not even close. What he had in mind was to assign a physical property to a dimension. A Hilbert space isn't it.

I didn't intend to reply to the OP initially since there's a lot of misconception in it (theoretical physics is nothing but NUMBERS?) and it'll take a lot of time to respond to all of them. However, I will address one thing that we ALL know if we have done any significant work in physics - that increasing the number of "dimensions" often increases the complexities of the problem. Think of what happened when you add another degree of freedom to the system that you're trying to solve. The dynamics either get more complicated, or even unsolvable in closed form. Anyone doing Lagrangian/Hamiltonian mechanics would have seen that repeatedly.

And really, this has nothing to do with QM. If one can make the problem easier by putting all the physical quantities into their separate "dimension", we could do this with classical mechanics also. Would anyone gain any more insight into the problem by putting all those physics quantites into their own dimensions (whatever that means)? I don't think so.

Having said that, we often deal with phase space in which, for example, we have 3 spatial and 3 momentum space, giving us a 6-D phase space. But this is nowhere near what the OP intended and it isn't used as a "theoretical simplification" (6D phase space is horrible to visualize and it does nothing to help us solve typical solid-state problems).

Zz.

 

[nameta9]

Thanks a lot for your reply zapper. It is just an idea that I had, that seemed intriguing. The fact that each property can be assigned a dimension may be used in computer simulations and could be visualized through code. I don't know if any research like this has ever been made. Maybe some smart theorists will take my idea and make the next breakthrough discovery.

This idea also does away with any residual metaphysics since mass no longer exists, and neither does spin or energy or all the other metaphysical inventions we made to understand physics. We now have pure numbers and geometry.

 

[ZapperZ]

Thanks a lot for your reply zapper. It is just an idea that I had, that seemed intriguing. The fact that each property can be assigned a dimension may be used in computer simulations and could be visualized through code. I don't know if any research like this has ever been made. Maybe some smart theorists will take my idea and make the next breakthrough discovery.

This idea also does away with any residual metaphysics since mass no longer exists, and neither does spin or energy or all the other metaphysical inventions we made to understand physics. We now have pure numbers and geometry.

What you need to do is show that this WILL work, and WILL simplify things. You can't do this via hand-waving arguments. Start with a simple classical systems. And take note that the "dimensions" may not be orthorgonal to each other. How do you know the "mass" of an object doesn't couple of the spatial dimension? How do you know the "energy" of an object doesn't couple to its charge?

In the end, all I can think of here is that what you're doing is nothing more than "graphing". I do not see how this simplifies anything. Note that you can't turn "mass" or "spin" or "charge" to a SPATIAL dimension simply by putting it on an axis and then labelling them as not existing or needed. You need to show the PHYSICS on why doing such a thing is valid in the first place. I have seen no such arguments.

Zz.

 

[Berislav]

Actually, it's not even close. What he had in mind was to assign a physical property to a dimension. A Hilbert space isn't it.

Aha. Well, in that case he's wrong. EDIT: Or maybe I am wrong for referring him to Hilbert formalism. Since this is the Quantum physics forum I assumed that it would be that best thing to mention.

However, I will address one thing that we ALL know if we have done any significant work in physics - that increasing the number of "dimensions" often increases the complexities of the problem. Think of what happened when you add another degree of freedom to the system that you're trying to solve.

What about gauge theories?

Having said that, we often deal with phase space in which, for example, we have 3 spatial and 3 momentum space, giving us a 6-D phase space. But this is nowhere near what the OP intended and it isn't used as a "theoretical simplification" (6D phase space is horrible to visualize and it does nothing to help us solve typical solid-state problems).

Phase space can be mathematically useful. We can define a 2-form on it and then we can treat it like a (symplectic) manifold.

 

[ZapperZ]

What about gauge theories?

And what about string, superstring, M-theory having all those "spatial dimensions"?

At some point, the problem being solved is no longer tractable. You'll see that the ONLY way to make any progress is to increase the number of dimensions. It's a way out via renormalization in many cases. However, you don't do this to make the problem simpler, as claimed in the OP. Note that what is being done even here is again not what the OP wants to do.

Zz.

 

[reilly]

No matter how you represent a physics problem, you've got to solve it. Sure, it's basically trivial to represent each independent variable as an independent dimension. But, then, what do you do?

If theoretical physics is just a bunch of numbers, then written language is just a bunch of letters, and you are a bunch of molecules. Where do you go from here?

Regards,
Reilly Atkinson

 

[beautiful1]

Hi nameta9,
This discussion reminded me of a passage from Landau and Lif****z's Statistical Physics, which I reproduce below for those interested:

In principle, we can obtain complete information concerning the motion of a mechanical system by constructing and integrating the equations of motion of the system, which are equal to its degree of freedom. But if we are concerned with a system which, though it obeys the laws of classical mechanics, has a very large number of degrees of freedom, the actual application of the methods of mechanics involves the necessity of setting up and solving the same number of differential equations, which is in general impracticable. It should be emphasized that, even if we could integrate these equations in a general form, it would be completely impossible to substitute in the general solution the initial conditions for the velocities and co-ordinates of the particles, if only because of the amount of time and paper that would be needed.

Statistical Physics (2nd Ed.), Second paragraph of Chapter 1.

I do not mean this to discourage you from your theoretical pursuits, but rather I mean to agree with reilly's point that the usefulness of a theory is determined by what results it can produce.