Alternative quantum theory

[Archaea]

I was originally going to post this in the thread: An alternative derivation of special relativity. But this isn't really relativity and that thread's a mess, so I thought I'd start this one.

This isn't really an alternative to the whole of quantum theory. What I've been looking for is a way to rigorously derive the Hamiltonian used in Schrödinger's equation. However, because the Schrödinger equation is so fundamental to much of quantum theory, changes to it mean there many changes to the theory.

The first thing to do is to take out the minus sign in the momentum operator in the position representation and put it into the position operator in the momentum representation:

P = iћ(d/dx) and x = -iћ(d/dp)

This has the effect of changing the signs of the commutators, so: [x, P] = -iћ and [P, x] = iћ. This needs to be done in order to make the commutators of the Hamiltonian work. This also means that the momentum eigenstates take the form e-ikx and the position eigenstates are eirp which shows up in the Fourier transforms.

The second thing we need to do is define:

F = dP(t)/dt and v = dx(t)/dt

These are supposed to be relativistic variables and not operators, so they are in need of a precise definition and the statement that they're variables needs to be justified... I think: The momentum eigenstates ψk(x) and ψj(x) will have a definite momenta pk = ћk and pj = ћj respectively. In the presence of the same "force" these eigenstates will have their momenta changing at a constant rate, so that:

dk/dt = dj/dt

And since a wave function can be decomposed into eigenstates, F will not depend on the wave function itself and so doesn't need to be an operator. We can do the same thing with v using position eigenstates.

The next thing to do is define the Hamiltonian as:

Ht = iћ(d/dt)

And look at the commutators in the position representation:

[Ht, P] = -ћ2[d/dt, d/dx] = 0
[Ht, x] = iћ(dx/dt) = iћv

And the commutators in the momentum representation:

[Ht, x] = -ћ2[d/dt, d/dp] = 0
[Ht, P] = iћ(dP/dt) = iћF

It seems to me that in quantum mechanics if something commutes with the right hand side then it commutes with the left hand side (except in the mainstream Schrödinger equation). With this in mind we can create two separate Hamiltonians for the position representation and the momentum representation:

Hx = vP
HP = -Fx

This is a bit funny, but after a bit of thought I decided it was okay, my thinking is that potential energy doesn't have any effect in the position representation and kinetic energy doesn't have any effect in the momentum representation. Also it makes sense not to add x and P together in the same equation because they don't commute.

Both the equations above are trivially correct due to the chain rule. The Hamiltonian equations of motion are satisfied as well:

F = -(dHP/dx)
v = (dHx/dP)

Finally, if we turn these into differential equations and cancel out the i's and the ћ's we get:

And combine these with the Fourier transforms (which I hope I got right):

I don't know if there's a way to get these into single differential equations for ψ(x) and φ(p) as I haven't tried yet. These equations are relativistic equations, which is good. The big problem I'm having now is that trying to find solutions gives weird answers, so I think it's a nice theory, but it might not give nice predictions. :)

 

[Archaea]

The second differential equation is wrong in my last post, it should be:

The solutions to these equations can be any function which takes the form:

For the position representation, cn/bn = v for all n . Since the velocity of a wave is ω/k we can make bn = kn and cn = ωn, and set a = -i. For the momentum representation F = E/-x, canceling out the ћ's gives F = ωn/-rn, so we can just make bn = -rn and cn = ωn, and set a = i.

Since these solutions are Fourier series, I think it's OK to generalize to Fourier transforms, which makes the solutions:

Where k, r and ω are functions (I can't for the life of me figure out what they're functions of, however) where ω/k = v and ω/-r = F.

So the solutions to the equations are the Fourier transforms... If I got everything right... This isn't as good as being able to definitively find the wavefunctions from the potential energy, but I think it makes sense. If you think of the wavefunction as being like an EM signal, where the signal can be anything it pleases, then the kinetic and potential energy are just going to determine the way the signal changes in space/momentum space and evolves in time.

 

[Archaea]

I forgot to put in the n's in the second equation, Jeez life is rough.

 

[Inquorate]

Were there equations here? They're not showing up

 

[Nienna]

Were there equations here? They're not showing up

I can see some equations, not that I understand any of it.

 

[Zasino]

Were there equations here? They're not showing up

I can see some equations, not that I understand any of it.

Archaea, the pictures in your two next-to-last posts are not showing in regular reading mode, although I can see them now while writing this post. funny.

Edit: ok never mind, they are all visible now.

 

[Archaea]

I'm thinking that from now on I'll write Hx as T and Hp as V. They're both the same operator, more or less, it's just that T operates in the position representation, while V operates in the momentum representation.

So there's a funny thing with this stuff. If we differentiate vP with respect to p, and set v = dT/dp we get:

v = dvP/dp = (dv/dp)P + v(dP/dp)

So dv/dp = 0. We can do this a better way:

dTψ/dp = (dT/dp)ψ + T(dψ/dp) = d(vpψ)/dp = (dv/dp)pψ + v(dp/dp)ψ + vp(dψ/dp)

And again dv/dp must be 0. This means that v doesn't vary as p varies. We can do a similar thing in the momentum representation, if we differentiate -Fx with respect to x, and set F = -dV/dx then we get dF/dx = 0. This means that F doesn't vary as x varies.

So either this theory is garbage or it's time to start making stuff up.

The obvious way to fix this is to make v = c, where c is the speed of light, and make F the F in gravity. This works because c is supposed to be constant in all inertial reference frames, which means c doesn't vary with momentum, and the force exerted by a gravitational field doesn't depend on the distance from the source, so F doesn't vary with distance.

An interesting effect of doing this is that we can square the equation T = vP to get:

T2 = v2P2

This is the electromagnetic wave equation. In fact we can raise the equation to any power and the complex exponential will be a solution to all of them. If this is true then the electric and magnetic fields of an electromagnetic wave are proportional to the wavefunction:

E(r, t) ∝ ψ(r, t)
B(r, t) ∝ ψ(r, t)

Since I'm just making stuff up, what I'm going to do is say that most of the basics in quantum mechanics refer to light (EM waves/photons) and most of the advanced quantum mechanics stuff is just wrong. So the normalization condition means that the amplitude of an EM wave will go to 0 as the wave goes to infinity. And an EM wave is most likely to collapse into a photon at the points where it's electric and magnetic fields are the highest/lowest.

 

[Archaea]

I was a bit confused in my last post. Since the solutions to the differential equation are complex exponentials and not ordinary sine waves, the probability of an EM wave collapsing into a photon at some point is the same at every point. Also the force of gravity is proportional to the inverse square of the distance, so the F isn't the F in gravity.

Instead, what I think is that if an observer observes the the wavefunction (EM wave) as traveling at the speed of light (c) regardless of the wavefunctions momentum (p) or the observer's inertial reference frame, then the same observer in a field of force will see the same shift in the frequency/frequencies of the EM wave regardless of the wavefunctions position (x) or the position of the observer.

I think this is OK even though the force might be higher in one place than another because if the only force that effects light is gravity, then time should be running slower at the points where the force is higher.

 

[Archaea]

So far this theory is just a one dimensional theory, where the direction of the EM wave or light ray has been along the r axis, say. Extending this to 3 dimensions should be pretty straight forward. If we define operators Tx, Ty, and Tz as well as operators Vx, Vy, and Vz so that:

Tx = (∂x/∂t)Px	Ty = (∂y/∂t)Py	Tz = (∂z/∂t)Pz
Vx = (∂px/∂t)x	Vy = (∂py/∂t)y	Vz = (∂pz/∂t)z

Note that the derivatives in the above equations can't vary with their corresponding operators.

The total kinetic and potential operators should be:

T = Tx + Ty + Tz
V = Vx + Vy + Vz

I think that for studying crystals (which I want to do to as a way of testing this theory without having to run any actual experiments) it might be easier to just use the Ti/Vi (where i = x, y or z) operators instead of the total energy operators, because crystals can have different symmetries along different axes.


If the z axis angular momentum operator is:

Lz = iћ(d/dθ)

Where the minus sign is gone because of the changes made to the momentum and position operators at the beginning. Then we can just multiply it by dθ/dt to get the total kinetic energy operator. We can also multiply Lz by dθ/dr to get Pr which gives T = (dθ/dr)cLz.

Also since the minus sign has been taken out of the momentum operator the commutators for the angular momentum operators are now:

[Lx, Ly] = -iћLz
[Ly, Lz] = -iћLx
[Lz, Lx] = -iћLy

 

[Archaea]

I've been having a bit of trouble trying to figure out what to do next with regard to developing this theory. The problem is that the Hamiltonian in classical physics is a function of phase space, but due to the uncertainty principle, phase space isn't well defined in quantum mechanics. This means that the kinetic and potential energy operators of this theory aren't really Hamiltonians, so the equations linking the Hamiltonian function and the Lagrangian function aren't applicable. The action isn't well defined either for a similar reason.

This means that the ideas and mathematics used in QFT and QED aren't applicable to this theory either. Although I like Professor Feynman's idea of path integrals and time slices and I think that these ideas might well have a place in some correct theory. However, I also think that a "path" isn't a well defined concept in quantum mechanics, and as far as I can tell it would have to be defined as a set of position eigenstates, then perhaps something could be done with space-time slices, but that would be way down the track.


Anyway, here are a couple of interesting Wikipedia pages: Arc length and Metric tensor. if we define a matrix J with elements Jab = ∂xa/∂x'b and a vector of operators Pa = iћ(∂/∂xa) then by the chain rule we get:

P'b = JPa

If we pretty much just copy what was done in this section of the metric tensor Wikipedia page we get:

PTbgPb = P'TaJTgJP'a
= P'Tag'P'a

I hope that's right, if we then hit this with a wave function and multiply by c, we can find the square of the kinetic energy of a wavefunction on a surface. We can do a similar thing in the momentum representation to find the square of the potential energy.

That's all a bit messy I think, but if we could say something like the kinetic energy is invariant under a change in position or the potential energy is invariant under a change in momentum then it might be possible to say something about the metric tensor g, and the types of possible surfaces.


Another thing about this theory is that since it's just a theory about electromagnetism and photons, then it suggests that at small scales every magnetic field has an associated electric field and vice versa, i.e. they can't be separated. This means that an iron bar magnet should have a small electric charge at it surface, since for large configurations it's possible for all the electric fields to cancel while all the magnetic field combine. However I should point out that I'm assuming EM waves can spin on or around a point.

 

[John G]

You could perhaps put your Feynman paths on a Feynman Checkerboard and handle the uncertainty principle via splitting & merging virtual (vacuum) worldlines between Checkerboard world states. A random walk on a Checkerboard gives the Dirac equation. The full symmetry group for the Dirac equation is the conformal group so this kind of matches with your relativity discussion. Perhaps you could use symmetry groups from a classical model for the QFT too. A conformal metric would be 4 spacelike and 2 timelike but there could be even more Kaluza Klein-like dimensions to handle the Standard Model forces. You can also think of the conformal metric as 4 complex dimensions or 4x4 position x momentum Dirac gamma matrices or 4x4 matrices from 4 coincident Checkerboards. Stick all this plus the fermion creation/annihilation operators in your QFT and your symmetry group gets quite huge.

 

[Archaea]

You could perhaps put your Feynman paths on a Feynman Checkerboard and handle the uncertainty principle via splitting & merging virtual (vacuum) worldlines between Checkerboard world states. A random walk on a Checkerboard gives the Dirac equation. The full symmetry group for the Dirac equation is the conformal group so this kind of matches with your relativity discussion. Perhaps you could use symmetry groups from a classical model for the QFT too. A conformal metric would be 4 spacelike and 2 timelike but there could be even more Kaluza Klein-like dimensions to handle the Standard Model forces. You can also think of the conformal metric as 4 complex dimensions or 4x4 position x momentum Dirac gamma matrices or 4x4 matrices from 4 coincident Checkerboards. Stick all this plus the fermion creation/annihilation operators in your QFT and your symmetry group gets quite huge.

I had a look at the Wikipedia page for the Dirac equation and Feynman checkerboard. I didn't know these two things existed, so thanks for pointing them out :). The original Dirac equation:

Looks the same to me as the equation in the theory I'm working on. However, I think the first term with the mc2 in it unbalances the equation, and I'm still trying to figure out what spin is, so I'm not sure about having matrices in the equation.

I wonder though whether we can get this equation :

From a Feynman Checkerboard for a spin-less and massless particle?

 

[John G]

On a single Feynman Checkerboard, a single massless (with no spin/charge) particle from its reference frame point of view would never see a change in time so it fits with special relativity in general. You would have to have the virtual (vacuum) bosons to get it from one checkerboard world state to another for any time change.

For what that looks like, in gr-qc/9910099, Chris Doran says: "... A new form of the Kerr solution... is global and involves a time coordinate which represents the local proper time for free-falling observers on a set of simple trajectories. ... The Kerr solution ... is global, making it suitable for studying processes near the horizon. ... the time coordinate measured by a family of free-falling observers brings the Dirac equation into Hamiltonian form ... This form of the equations also permits many techniques from quantum field theory to be carried over to a gravitational background with little modification. ... ".

Back to me: So you have a Kerr-Newman metric which for a massless no spin/charge particle has the particle seeing spacetime as four complex dimensions aka requiring the conformal group.

http://vixra.org/pdf/1303.0166v4.pdf

At Temperature / Energy above 3 x 10^15 K = 300 GeV: the Higgs mechanism is not in effect so there is full ElectroWeak Symmetry and no particles have any mass from the Higgs. Questions arise... What do physical phenomena look like in the Massless Phase ? Two points of view are important: ElectroWeak Particles and Dark Energy Conformal Gravity... Bound structures (Hadrons, Mesons, Nuclei, Atoms, etc) are based on standing wave frequencies instead of masses of particles, nuclei, etc... The geometry of the Conformal Sector is closely related to the Penrose
Paradise of Twistors. Yu. Manin in his 1981 book "Mathematics and Physics" said: "... In a world of light there are neither points nor moments of time; beings woven from light would live "nowhere" and "nowhen" ... the whole life history of a free photon [is] the smallest "event" that can happen to light. ...".

back to me: so things get a little weird.

For the matrices, I certainly don't get very much into the math but I tend to think of it as a Triality of Feynman slash notation matrices for n-component spinor fermions, n-component spinor antifermions, and n-momentum. Being a Triality, I tend to think of n as 8 which would be 4 extra Kaluza Klein dimensions.

 

[Archaea]

Found two Wikipedia pages: Kerr metric and Kerr–Newman metric. I'm not quite up to GR yet, and I think ultimately symmetry groups will be part of the theory, but what I'm trying to do now is take other people's ideas and make a mathematically consistent theory. ATM I think spin and mass are less fundamental than electromagnetism, but having group matrices in the Dirac equation implies to me that spin would be more fundamental.

I think theoretical physics has a lot of unbalanced equations, it seems to me that this is a result of having to put scientific hypotheses into mathematical form, as well as a conspiracy to brainwash physics students. Ultimately I think the mathematical side of physics needs to be a fairly dry subject, and new hypotheses need to formed to be mathematically consistent with old hypotheses or the old hypotheses need to be shown to be false.

 

[John G]

The matrices are really creation/annihilation operators via spacetime components x spinor fermion/antifermion and position/momentum operators via spacetime position x spacetime momentum. In Wikipedia's Dirac equation article it says:

The Dirac equation may now be interpreted as an eigenvalue equation, where the rest mass is proportional to an eigenvalue of the 4-momentum operator, the proportionality constant being the speed of light:

...in Feynman slash notation, which includes the gamma matrices as well as a summation over the spinor components in the derivative itself, the Dirac equation becomes:

back to me: The idea of spin via spacetime can be seen here (the Feynman Checkerboard is spin network-like):

https://en.wikipedia.org/wiki/Spin_network

So yes I would agree mass and spin are not fundamental.