April 10, 2003
Philosophically, the highest success of physics in explaining the fundamental nature of physical reality has to be relativity, which showed that space and time are part of the same object, and gave the rules for converting between them. The incredible successes of quantum mechanics in calculating everything from particle lifetimes to cross sections has not been associated with an equivalent improvement in our understanding of fundamental reality. The reason for this is fairly clear; after most of a century, the interpretations of quantum mechanics are not only still unintuitive, they remain subject to debate. While the nature of the reality behind the experiments is still not just cloudy, but completely opaque, the calculations for those experiments have been markedly accurate. Quantum mechanics has been successful at everything except at being understood.
These difficulties are well known to physicists (who are the intended audience of this note). I'm including them so that I can give my particular spin on the interpretation of these conundrums, as well as to bring them to mind.
Bohmian mechanics is an alternative interpretation of quantum mechanics where additional variables are added. The added variables indicate the actual positions of the particles. That is, the particles are assumed to always have precise locations. In order to account for interferences, an additional potential ("quantum potential") is added to the usual (classic) potential. Thus a complete description of a particle is its wave function and it's initial position. Note that the wave function does not depend on the initial position, but the particle track depends on both the initial position and the wave function. This is the most (only?) successful hidden variables theory.
Surprisingly, if you assume that the input states to an experiment follow the usual quantum mechanical distribution, Bohmian mechanics shows that the output states will also follow those distributions. That is, under the assumption that the universe is already follows the usual quantum mechanical distributions, Bohmian mechanics has predictions for the evolution of those distributions that are identical to those of quantum mechanics. In short, Bohmian mechanics reduces quantum mechanics into a branch of statistical mechanics. Maybe it would be the standard theory if it didn't have a few problems.
Bohmian Mechanics' most difficult problem is said to be that it does not yet have a field theory. A recent paper by S. Goldstein, D. Dürr, R. Tumulka, and N. Zanghì "Bohmian Mechanics and Quantum Field Theory" gives a simple field theory for Bohmian mechanics, but it has the same disadvantage that regular quantum mechanics has, the particle interactions are handled probablistically. Sheldon Goldstein's articles are linked from his web site here: http://www.math.rutgers.edu/~oldstein/ But if Bohmian mechanics itself is any guide, a deeper understanding of particle interactions (maybe in the form of a wave function) will provide an opportunity to replace these probablities with calculated results from initial conditions. So from the point of view of understanding the physical nature of the world, I don't find the lack of a field theory to be decisive, per se. What is more important is that Bohmian mechanics is not relativistic (but see paper cited above).
A minor problem with Bohmian mechanics is that it implies that the particle trajectories that it computes turn out to be stationary for some bound states. In particular, the "s" states for a central force are purely real, and so have no probability flow. These states (and others) therefore have no electron movement. Of course this is in a playpen version of the theory, and one presumes that an analysis that included a dynamic vacuum would add enough randomness to walk the electron around the nucleus.
And I have a philosophical problem with Bohmian mechanics in that it seems odd that the universe would associate such diverse things as an exact particle position with a wave. For this I have a suggestion, a way of combining these two disparate notions. I'll discuss it in my notes on Schroedinger's equation, but the basic idea is to extend Schroedinger's equation to a single (simple) equation that includes both particle and wave solutions.
Symmetries are very attractive to the human brain, and there may be a tendency to exaggerate their importance. Certainly the periodic table of the elements has many beautiful symmetries, but it would be difficult to derive Schroedinger's wave equation from those symmetries. Instead, one would have to happen upon the correct wave equation, and then notice that its solutions possessed the symmetry one observed.
Symmetries that are apparent at our usual energies may be accidents that obfuscate the underlying simplicity. In a contest between perfect simplicity and perfect symmetry, the place to put your bet is on simplicity. For computing matrix elements, imperfect symmetries are great, but the fact that they are not perfect is an indication that they are probably coincidences, rather than indications of the fundamental reality.
Historically, most of the advances in physics have come from rejecting symmetries that were previously assumed clear. The earth was once considered the center of the universe, an obvious symmetry. The planets were assumed to go around the sun in circular (S1) orbits, or in orbits defined by the meshing of spheres. Time and distance measurements were assumed to be everywhere equal. If a Lagrangian possessed a certain symmetry, then it was assumed that its vacuum states would too. Etc.
The simplicity of the mathematics will imply that it may not be very useful as far as predicting the results of high energy particle experiments. This would be consistent with our present advances in physics. For example, while QED has been very successful at improving calculations for high energy particles, it has not helped improve calculations in more prosaic physics such as band gaps in germanium semiconductors. A new dynamics for extremely high energy situations needs to provide a framework where lower energy dynamics are explained, but it need not improve the calculations for those lower energy situations.
When relativity first appeared nearly 100 years ago, it did not improve calculations for much of physics. It was only with the passage of time that it worked its way into the wide range of physics where it is visible today. We can't expect a new dynamics theory to be a super theory that explains everything, but instead we should expect it to be a simple tool that explains a few of our outstanding mysteries.
I should mention that there is a way of replacing the complex numbers from Schroedinger's equation in a way that produces a simpler equation. The idea is to replace factors of i in the equation with d/ds, where d/ds is a derivative in an extra dimension, and simultaneously add a factor exp(i s) to the wave function. The resulting equation supports all the usual solutions of Schroedinger's equation, but also includes extra solutions which, depending on various choices, correspond to particles with higher mass or coupling constant. I'll include more details further on in this note.
By banning complex functions, I don't mean to exclude them where they simplify the mathematics, only to distrust them as far as their being fundamental objects. For example, physicists commonly use exp(it) for a sinusoidal function, taking real parts at the end of the calculation.
One of the problems with formalism is that they almost always assume the existence of symmetries. By using formal theories, one ends up being unable to produce a theory which violates those symmetries, even if the violation would be below the level detectable with current experiments.
Noether's theorem, which associates symmetries with conserved quantities, is a good example of the attractiveness of symmetries. The Heisenberg uncertainty relationship suggests that energy is only conserved on a probabilistic basis. A theory which failed to conserve energy (or momentum), in some small amount, would therefore not only be consistent with observational evidence, but the existence of that uncertainty relationship is almost equivalent to a blinking neon sign: "look over here, no one else did." The universe is limited in terms of its age and size, so Heisenberg's uncertainty principle places limits on how accurate it is even theoretically possible to design an experiment intended to test for conservation of energy and or momentum. In other words, not only are energy and momentum not known to be exactly conserved, present theory holds that they are not, in fact, conserved exactly at all. Under this situation, would it not be natural that a fundamental theory would have conservation of these things as an approximate symmetry rather than an exact one? Note: I don't propose to repeal these conservation laws, I am making this point only to give the professional reader a reason to briefly consider theories not written in the formalism of Lagrange or Hamilton, and perhaps not writable at all that way.
This can only be accomplished if our usual understanding of either particles or time is modified. Before the particle actually makes its voyage its wave function is aware of all the choices, but as time goes on, the wave function collapses on to the path actually chosen. The current standard interpretation of Bohmian mechanics makes this possible by by defining the particle to have two parts, a wave function and a position. This is more complicated than I'd like. I think that there is another way of accomplishing the same thing, and that is to consider the position of the particle as defined by a wave collapse that occurs as proper time transpires. This leaves the particle able to be influenced over space-like separated regions before the wave is collapsed, and as time passes, the particle's wave is collapsed down to its position. Long after the experiment has completed, we can observe the particle's track, but we can also see the influences of the wave nature of the particle. The result is that the particle's wave did traverse both slits, but as time progressed, the particle's path became frozen and a particular path was selected.
Adding a decay rate that is proportional to proper time does add complexity, but it may be that there is a simple underlying equation that can be interpreted to have a standard quantum mechanical wave part for the time before the particle goes through, and also a decay to a collapsed wave function afterwards. Certainly it's no worse than having things as they now stand, with waves and particles having to share the fundamental spotlight.
Note that proper time is associated with individual particles. It is not an attribute of a reference frame or a collection of particles, as each of those particles may have different ages.
For reference, Schroedinger's wave equation for a scalar particle in a potential V(r):
h ∂Ψ / ∂ t =
(- h / 2m Δ + V(r)) Ψ
Before going into further discussion, I should first note that this equation applies only to experiments that have not yet been run. That is, it is a postulate of quantum mechanics that after you run an experiment, the wave function is collapsed into whatever you end up observing. In that sense, the t variable in the above equation can only be interpreted as applying to a time in the future, not the past.
Over the years more than one mathematician (including my father who bought the copy of Messiah quoted later on) has asked me more than once why it is that the wave function is complex and requires an absolute value and squaring operation before it can be interpreted as a probability. I think this is a useful question to address.
Before a student can understand (standard) quantum mechanics, he must first study classical mechanics. Classical mechanics originated with analyzing a collection of masses, with each mass having a position and velocity (or momentum). These are governed by Newton's three laws, which are still taught. In order to more simply analyze situations where particles are constrained (by ropes, for example), and/or where it is desired to use more arbitrary coordinates (spherical coordinates for central forces problems, for example), Lagrange's equations of motion are used. In these 2nd order equations, the positions and velocities become "generalized coordinates" and "generalized velocities" and the constraint equations are no longer apparent but are implicit. The Lagrangian equations of motion can be transformed to Hamiltonian equations, which replace the 2nd order differential equations with twice as many 1st order equations. It is perhaps indicative of a certain mindset in theoretical physics, that the variables now containing the information about positions and velocities are known in the Hamiltonian formalism as "canonical coordinates".
The Lagrangian or Hamiltonian formalism can be naturally extended to continuous systems of matter. It is this continuous extension that is directly connected to quantum mechanics.
The Hamiltonian formalism for continuous systems requires two functions to describe the field. That there are two functions is why the quantum wave state cannot be described with a single real valued function (though I will later in this chapter show an extension of Schroedinger's equation that does require only a single function). Having a single real valued function could provide a probabilty density, but it would not also be able to provide a velocity at the same time. Yes, I know that this more or less directly contradicts my previous sentence, but to get a velocity out of a real valued function I have to add an extra dimension.
Note that the Hamiltonian coordinates are generalized coordinates. This means that there is no particular reason why they should correspond to a probability density as opposed to a complex square root of a probability density. In fact, it's easy enough to rewrite Schroedinger's wave equation so that one of the two wave functions is the probability density (as we will see in the next section). These observations suggest that the question of "why does the quantum wave function require squaring a complex absolute value to get a probability density" could instead be written: "why is the complex version of the quantum wave function so simple?".
One of the most fruitful ideas of physics has been the principle of linear combinations. Sine waves are particularly useful in physics because they can be added together in linear combinations to create more sine (or cosine) waves. Another very fruitful assumption in physics has been the principle of perturbations. The combination of these fruitful assumptions suggests that a simple theory of quantum mechanics would arise from equations which (a) support sine waves, and (b) are linear. But sine waves go negative, so they can't be used as probability densities. The simplest way to convert such a wave to a legal probability density is to square it.
Given the above, and recognizing the formal nature of quantum theory, it is natural that the quantum wave state be given by a complex valued function. But one might also ask if this is a case of the missing wallet being searched for under the lamppost. The nonlinear, non perturbed parts of physics are the more complicated ones, so it is natural that our dearest successes have been in the areas where linear combinations and perturbation analysis work best. This is not in the region of time where the wave function has collapsed and is concentrated, but instead in the future, where the wave function is as dilute as possible.
There is another clue from the above notes, and that is that the formalism of quantum mechanics is based on principles of classical mechanics that involve constraints. This suggests that when we look for more fundamental descriptions of quantum mechanics that involve extra dimensions, we look around for more or less natural constraints that can be added at the same time. In particular, what kind of constraint would turn a simple version of Schroedinger's equation (using the probability density and velocity field version) into a simple but less linear equation perhaps using more dimensions?
Rather than convert all the gory details into HTML, let me quote Messiah who does pretty much
what we want, but uses
Ψ = A exp(i S/
and then goes on to take the classical approximation h -> 0. My copy is apparently the
first English edition (John Wiley ~1961), look in pages 222-224 of volume 1, in the beginning of the
section entitled "Classical Limit of the Schroedinger's Equation".
Making this substitution, and seperating into real and imaginary parts, and multiplying
by 2A gives two equations, one for ∂S/∂t
[refer (VI.17) in Messiah], the other for ∂A/∂t [refer (VI.19)]:
(5.2a) ∂S/∂t + grad2 (S)
/(2 m) + V
h2 / 2m) (Δ A) / A,
(5.2b) m ∂(A2)/∂t + div(A2 grad S) = 0.
Equation (5.2b) is simply the continuity equation for the probablity density A2. It can be interpreted in a purely classical manner. Messiah interprets S to be a potential that generates a velocity field v for that probability density (where J is the usual probability current density) [refer (VI.21)]:
(5.3) v = J/A2 = grad(S) / m.
Messiah then shows that (with
h = 0) the velocity field follows the law of motion for a
classical fluid under the influence of the given potential. But if he had not taken the classical
limit, the conclusion that S was a potential for a velocity field would have still been valid,
as there is no h in the probability continuity equation (5.2b).
Since the probability density acts like a classical fluid, as opposed to a gas, it need have only a single velocity defined at each point. This is not what a statistical ensemble of (non interacting) particles with various positions and velocities would give. Such a collection would be a gas, rather than a fluid. Instead, the "particles" strongly interact, and they interact in such a way as to allow only a single velocity at each point in space.
S only shows up with a gradient, except for the term indicating how it changes with time. It is therefore possible to take the gradient of that equation (5.2a), and then make a change of variables to replace grad S with m v:
(5.4) m ∂v/∂t
+ grad(v2 / 2m) + grad V
h2 / 2m) grad(Δ (A) / A).
As an aside, you can work the right hand side of the above equation to use A2 instead of A, but the result is about as complicated, and the final result can be written in several distinct ways. Also note that the only term that is not already a gradient is the ∂v/∂t term. This means that the above equation can be used to compute the time evolution of velocity fields which are not the gradient of any S.
Equation (5.4) is useful in that Schroedinger's equation has automatically built into it the assumption
that Δ X Δ P ≥
h. So if you wanted to see what the time
evolution of a wave state that was initially more tightly concentrated than Heisenberg's uncertainty
principle would allow, you cannot evolve the function using Schroedinger's wave equation. That is, if
you minimize ΔX by packing all of A2's support into a very small region,
you end up with ∂/∂x being very large.
But by using the above equation, you can initialize both the position and the velocities to whatever
you like (and use v explicitly for the probability density velocity). Of course if your
initial conditions satisfy Schroedinger's wave equation, then the computed evolution will match
Schroedinger's. Since all the changes to mv can be written as a
gradient, one would expect that the discrepancy between mv and the velocity field
that has a potential, and therefore that can be associated with Schroedinger's wave equation
would decrease with time as the effects of the initial conditions recede into the past, though
I have not derived this.
Equation (5.4) also suggests a way of combining the particle and wave in Bohmian mechanics, and thereby make more explicit the probability wave collapse. As I mentioned earlier, Schroedinger's wave equation is suitable only for experiments that are in the future. But equation (5.4), which includes Schroedinger's, also has solutions for waves with position and momentum both as accurately described as needed. This makes a wave function that is more compatible with our own intuitive nature of time. Only the future is nebulous, the past cannot be changed. But while we gloss over the transition between past and future, equation (5.4) allows that transition to be made explicit.
In this interpretation, the particle's position and momentum were perfectly (at least exponentially perfectly) defined in the far past. The present corresponds to a period where the error in position and momentum exponentially increase, until they knit into the future where Heisenberg's uncertainty principle applies completely. Equation (5.4) supports solutions to the wave equation with this property.
But there are problems with equation (5.4), chief among them being that it doesn't have the right statistics or spin. That is, when you translate the interference that we all know that quantum waves experience back into equation (5.4), you end up with a horribly complicated mess. So while (5.4) gives the feel for what I believe we should be searching for, it is not a very direct signpost to that spot. On the other hand, equation (5.4) is compatible with the usual quantum mechanical spin, and it supports solutions with SU(2) symmetry without the need for a complicated or arbitrary topology for space-time, or for functions with more than the usual classical complexity. More on this subject in a later section, where we will look at the classical interpretations of angular momentum and spin.
I should note that there are undoubtedly a lot of other equations that have Schroedinger's wave equation as an attractor. Maybe one of these is such a simple equation that it is a more natural candidate for a fundamental theory. But given the problems with a non relativistic spinless theory, I'm not inclined to spend a lot of time searching for one.
The first thing to note is that Ψ is a complex function, so its interference shows up in the form of a complex addition. The addition, when expressed in terms of our real variables A and S, is rather complicated. Why does it work so well? We all know where freshmen in college learn about adding complex numbers, but I fail to see who taught mother nature the details. The explanation that "mathematics models reality amazingly well" is true enough, but I'm looking for an understanding of physical reality, not a good model.
My guess is that Schroedinger's wave equation is a formalism rather than a fundamental part of nature. That is, the complex numbers are a symmetry that is not necessarily obvious in the underlying fundamental reality. More fundamental theories, like relativity, get by just fine with the reals, quantum mechanics should too. (Try finding a complex number in Misner Thorne and Wheeler's Gravitation.) Unfortunately, with quantum mechanics, Schroedinger's wave equation is the most fundamental object I've got, so any analysis has to start here.
Rather than looking at the 2-slit experiment, I think we will make better progress by looking instead at interference effects between two particles. The interference in the 2-slit experiment is understandable strictly as a wave effect, and as such isn't any more interesting than the similar effects that show up with light. I've already concluded that the fundamental nature of matter is waves, so the 2-slit experiment doesn't surprise.
But the interference effects between two (or more) identical or non identical particles are more interesting. There are four cases worth running through the calculations on, identical fermions (two electrons), identical bosons (two pions), distinguishable particles (an electron and a proton), and identical classical particles. Note that all this talk about indistinguishable particles is perfectly consistent with our thinking of them as some sort of waves. Of course waves are indistinguishable. If we instead thought of particles as point particles I would think that they would still be indistinguishable, but by assuming that the fundamental nature of them is a wave this doesn't come up. Waves is waves.
The quantum mechanical fundamental description for a single particle consists of its wave function, whose informational content includes a probability density and a velocity potential. The corresponding description for two particles is a joint probability density, and a joint velocity potential. The joint probability density is a function on R3 x R3, as it has to provide a density value for the two particles. The two position coordinates will be called x1 and x2. To be specific, let P(x1, x2) d3x1 d3x2 will be the probability that the the first particle is found within d3x1 of the position x1, and the second particle within d3x2 of the position x2.
I've already made the more sensitive wince, by refering to "first" and "second" particles just inches from having previously assumed that they are identical. Because of this defacto enumeration of the particles, most of our calculated probability density values will be 1/2 (for n particles it would be 1/n! ) the literal value, (but they'll be correct when x1 = x2). The reason physicists use this way of counting instead is that it makes the wave functions easier to normalize and calculate with. That is, if our space only had two points, a and b, then we will have P(a,a) + P(a,b) + P(b,a) + P(b,b) = 1.
We'll compare the four cases using notation common to the quantum case. So let P1(x) and P2(x) be the the probability densities (i.e. P = A2), and S1(x) and S2(x) be the velocity potentials for the corresponding quantum particles.
If the particles are distinguishable, the combined probability density is just the product of the densities:
(5.5) Pdist(x1, x2) = P1(x1) P2(x2)
If the particles are identical, but are classical, we have pretty much the same thing, but since the particles are identical, we have to take this fact into account. Note that the factor of 2 is from our splitting the density for x1 ≠ x2 into to two parts; the factor which may have caused wincing:
(5.6) Pclass(x1, x2) = (P1(x1) P2(x2) + P1(x2) P2(x1)) / 2.
The formulas for fermions and bosons are more complicated as they also depend on A. In these equations, the ± indicates bosons (+) or fermions (-):
(5.7) Pquant(x1, x2) =
(P1(x1) P2(x2) +
P1(x2) P2(x1)) / 2
± [P1(x1) P1(x2) P2(x1) P2(x2) ]1/2 cos [ ( S1(x1) - S1(x2) ) - ( S2(x1) - S2(x2) )/
The complexity of the interference term is consistent with quantum mechanics being a perturbational theory. That is, the theory is beautifully linear as long as no particles risk being put into the same position, but the interaction between two particles is highly non linear.
Note that S1 or S2 could have a constant added to it
with no change in the result. This is a global symmetry of S. But there is also
a local symmetry in that local changes to S by multiples of 2π
result in no
change in the joint probability density. These observations also apply to the joint
velocity potential, Squant(x1, x2).
Also note that if P1 and P2 share no support, then
the quantum mechanical formula is identical to the classical one, which supports the decision to divide
Ψ into two real functions.
The fact that the joint wave function does not depend, even locally, on changes in S by
suggests that S should not be considered to be a real function. Instead, it's natural range
should be S1. This explains, in a circular
manner, why it is that mapping of P and S into complex form results
in such a simple wave equation.
Also note that if one of the two wave functions has a constant added to S, there is no change to the resulting interference effects. That is, S1 only shows up as the difference in its value between two points. A global addition of a constant has no physical effect. This fact suggests that there is a simpler way of looking at spin than to represent it as an SU(2) symmetry.
From the discussion so far, one could get the impression that S can be a well defined real function of real space, but a brief consideration of bound states for the hydrogen atom with non zero angular momentum show that this is not the case. The simplest such state is the p-wave eigenfunction with angular momentum in the z direction of 1. Since life is short and HTML is long, I'm leaving off the normalization factors, and using spherical coordinates:
(5.8) Ψ211 = r exp(-r/2a0) sin(θ) exp(i φ).
Note that the above equation ignores the dependency on t, which I'll include back in later. For now it's sufficient to ignore time dependence, and to look at the interference effects at only a single moment in time. From equation (5.8), it's clear that S211 depends only on φ, and that it depends on φ in a discontinuous manner. That is, S is discontinuous at φ = 0 == 2π. This fact suggests that the natural way to implement Schroedinger's wave equation is with the v variable instead, using equation (5.4).
Since v is defined by the gradient of S, converting equation (5.4) into an equation for v instead of S requires an integral. Since the integral is a path integral, it's not obvious that the result does not depend on the path chosen. Of course if our solution is also a solution of Schroedinger's equation, then the existence of the velocity potential S implies that there is no path dependency. If we restrict ourselves to cases where x1 and x2 lie on the same trajectory, we can use the trajectory to define S from v without having this worry. But since Schroedinger's has this fact (that an S exists for any "physically possible" v) built into its equation, I'll ignore this problem for now. The resulting equation for quantum interference, as defined by velocity instead of velocity potential, is therefore as follows (with a very small amount of arithmetic), and I've chosen to parameterize the path integrals by η, with units of distance. This way the units are made obvious:
(5.9) Pquant(x1, x2) =
(P1(x1) P2(x2) +
P1(x2) P2(x1)) / 2
± [P1(x1) P1(x2) P2(x1) P2(x2) ]1/2 cos [ ∫12m (v1 - v2)/
h • dη ].
That (5.9) seems less complicated than one might expect is a good indication that we're not getting too far away from the trail of a simple equation with Schroedinger's wave equation as a limiting case. But about that cosine function. While a cosine is to be expected in any interference calculation, this interference is not the usual type. To contrast this interference from the type of interference we're used to in classical equations, it's useful to examine a classical interference, say between two cosines simply added together in the usual way. To simplify this a bit, let's take the two waves to both be cosines, both with unity amplitude. The more general case with various amplitudes of sines and cosines gives frequency results that are similar, but this case is sufficient to detail the interference.
(5.10) cos( A ) + cos( B ) = 2 cos((A + B)/2) cos((A - B)/2).
Equation (5.10) has the usual interference terms that we come to expect. Two frequencies are present in the combination. The first carries the average frequency and corresponds to a wave with an intermediate wave vector. The second depends on the difference, and is the inteference term.
Comparing to equation (5.9), we see that quantum interference is similar to classical interference, but only the A-B term is present. The two cosines on the left hand side of the equality in (5.10) do not show in (5.9), nor does the A+B term on the right hand side. The missing cosines are present only in the complex description of the wave function, they disappear when the wave function is split into the two classical parts. This is a little confusing, so let me explain further.
A quantum mechanical plane wave such as exp(i(kx-wt)) represents a steady flow of particles in a single direction, with the particle density everywhere equal. There is no sine or cosine dependence. Another way of describing this is to note that if one slit of the two slit experiment is covered up, the resulting distribution of particles on the screen is even both in space and in time. There is no physical significance to the oscillations in the quantum wave, except when the wave interferes with itself or others. Thus from a physical point of view there are three missing cosines/sines in the equation for quantum interference, as compared to classical interference.
So while quantum interference appears to be very distinct from classical interference, the absence of those three cosines may be due to a limitation of Schroedinger's equation. That is, the missing cosines may show up in a more fundamental theory, but are missing in Schroedinger's wave equation because their frequencies are too high. Instead, these very high frequencies may be averaged over in Schroedinger's equation, so only the low frequency interference term cos(A + B) appears.
So we go back and look at the interference term in equation (5.9) once more, and look for an alternative explanation for that interference, one more radical than classical wave interference. The explanation has to involve a structure that somehow combines velocities and positions, and it will probably result in either or both of those being quantized.
What's really odd about this interference is that it is completely cyclic in
h. That is, if you are considering points that are at a distance Δx,
an addition to the momentum mv1 of h /Δx results in
the same inteference. What's also significant is that you can
change the intervening momenta (that is, the momentum in the positions between x1 and
x2), and provided you change it by an amount that integrates out to 2π, there is
no way to detect the difference at the end
Of course it's an accepted fact that angular momentum is quantized. But the cosine
factor in equation (5.9) suggests that there is a milder form of a sort of quantization
for momentum itself. That is, that momentum integrated over a distance interferes with itself
according to multiples of
h. That this relation involves both
position and momentum suggests that it is an artifact of a deeper relationship between
It's frequently said that angular momentum in classical mechanics is analogous to quantum mechanical angular momentum, but that spin is a purely quantum effect. Spin adds the complexity of SU(2) symmetry to quantum mechanics, and it does it in a way that is difficult to extract. So to get to the fundamental physics of spin it is best to approach it first from the more convenient study of angular momentum.
One of the features of the half integer representations of SU(2) is that they require rotations by 4π in order to come back to return to the the unrotated situation. This feature shows up in spinor representations of fermions, and at first glance it suggests that the world is a very odd place. Classically, rotations by 2π leave the system unchanged, but with spinors, the wave function is negated.
Classically we can get along with just positions and velocities, can we achieve quantum spin with just these variables? Or do we have to follow the spinor representation and end up with a multiply valued fundamental representation of reality?
As was noted in section [5.b], changes to the wave function by multiplication by a constant do not change the physical situation. They correspond to the addition of a constant to S, and by equation (5.7), there is no resulting interference effect. Moreover, replacing Ψ with P and v completely removes the need for the spinor representations. It's easy enough to create a wave in P and v that has spin 1/2, in fact it's all too easy to create waves with perfectly arbitrary spin. Presumably the difference between particles of various spin is in the details of their wave functions. That is, while the Schroedinger wave equation enforces integer spins for (large) composite objects, it's possible that the equation is only a space averaging approximation for an equation that is applicable to smaller dimensions, and includes half integer spins available only in these smaller sizes. Later on in this paper, I'll show just such a space.
This all suggests that the R3 that makes up the obvious part of our real world is not the complete (local) topology of the world, but that there is at least one hidden dimension as well. This more complicated topology is going to have to be consistent with relativity, and support the odd quantum interference effects as well. Since we understand the principles of relativity a hell of a lot better than those of quantum mechanics, the place to start looking for this alternative topology is in relativity. Eventually it would be nice to include general relativity and gravitation, but special relativity definitely comes first.
As discussed in section , the fundamental time of the universe appears to be the proper time that is distinct for each individual particle. That is, the global time that we use in our equations seems to have the character of a convenient parameter to integrate over rather than a fundamental part of the world. So our modified topology should include proper time. And as integration is still a useful thing, we'll include global time as well. This suggests that we begin with the standard Lorentz metric, which allows the computation of proper time in terms of position and global time:
(6.1) ds2 = dt2 – (dx2 + dy2 + dz2).
While the abover formula is convenient for computing proper time, it doesn't show proper time as a independent dimension, so let's imitate Einstein a bit and instead treat proper time as if it were a spatial dimension. That is, relativity shows that space and global time are mixed by boosts. Our topology will instead mix space and proper time, and leave global time for calculational purposes only. The metric that we'll use is therefore:
(6.2) dt2 = ds2 + dx2 + dy2 + dz2
The above relation gives the amount of global coordinate time required by a particle to move (dx, dy, dz) given that it experiences a proper time interval of ds. It has the advantage over (6.1) that it is positive. So the local topology, instead of being Minkowski, will instead be Riemannian. This is a simpler topology than that of (6.1), so we can expect that some of the odd features of the usual topology of special relativity will vanish from the local topology, and will instead show up only in the global topology.
With the local topology selected (i.e. equation (6.2) which gives R4), we must now choose the global topology. We have proper time as one of the coordinates (or dimensions), but we have some interpretation issues. Different particles can experience proper time at different rates, but can nevertheless end up at the same positions. So the proper time dimension must be one that is ignored from the point of view of comparing positions. That is, to compare two positions (to see if there is a collision, for example), we must ignore the proper time coordinate.
Giving the proper time a cyclic coordinate will allow different proper times to be associated with the same (x, y, z) position. More formally, choose S1 as the topology of the proper time dimension, so that the full topology for points in the space such as (x, y, z, s) is R3xS1, with the S1 circle having a circumference of L. In this topology, the shortest (global) distance between two points will be influenced by the hidden dimension only by, at most, L/2, and so will be neglible for sufficiently small L. What's more, an error in distance of the maximum amount will occur only if the points occupy the same real coordinates.
The Proper Time topology violates the basic assumption of special relativity that all frames of reference are equivalent. That is, the topology defines a frame of reference that is the prefered one. But since this topology uses the same metric as Minkowski space, the fact that there is a prefered frame of reference may not be observable, but instead is as hidden as the hidden dimensions that allow this topology to function. Special relativity is an approximate symmetry of this topology instead of an exact symmetry, as will be shown later. For those who have trouble accepting that this topology does transform frames of reference the same way as special relativity does, nothing can be more convincing than calculations, which are provided in Appendix A.
This Proper Time topology is perhaps more complicated globally than Minkowski space,
but since the metric has no minus signs it is a standard differentiable manifold, and is
therefore considerably less complicated locally. The standard results of manifold theory
will hold in this space. This is exactly the kind of tradeoff
that we're looking for. That is, since the electron is (according to current
experimental data) a point particle, if we are looking for a version of Schroedinger's
wave equation that gives this particle its
h/2 spin, we must
add a complexity to the
local nature of space, but that complexity must disappear for larger distances.
In the Proper Time topology, the magnitude of the velocity of any particle is c (= 1).
(6.3) |( dx/dt, dy/dt, dz/dt, ds/dt)| = 1.
Another way of deriving the above relation is to note that if the velocity of a particle in the 3 real dimensions is β then by time dilation, the ratio of velocity in the proper time dimension is sqrt(1 - β 2), so the total velocity of the particle is 1. This relation is built into the metric chosen for the topology, so it should not be a surprise. In fact, you could do the same thing with any space with an arbitrary mechanics, provided the particle speeds have an upper bound. What makes this topology useful for our own universe is that proper time is such an important part of special relativity, and the relations that connect proper time to the regular dimensions make the metric particularly simple.
Having all particles travel at the same speed (at least locally), is convenient for calculational purposes, and we can expect that some of the relations of special relativity (such as conservation laws) will be simplified in this topology (and the simple derivations will be provided later). Again, this kind of simplification is what we want, and besides, since it is clear that matter is carried by waves, what would be more natural than to have all waves travel at the same speed? Waves, at least conceptually, require an "ether" to propagate in, and the most natural ether is one where all waves travel at the same speed. Anything else requires complicated dispersion relationships.
In the proper time topology, the proper time experienced by a particle is measured by how far it moves in the s dimension. While the s dimension is only a circle of circumference L, it is possible to go around that circle many times, so very long proper times can be represented by many orbits around that circle.
The defining metric for the proper time topology (6.2) is identical to the metric for standard special relativity, and so the two theories give the same answers when computing times and distances, (at least when ignoring very small discrepancies in positions), but for those readers that are unsure of the concept, or for those who want to see how time dilation and length contraction are computed in a proper time topology, as opposed to the global time topology of special relativity, I've included sample calculations for time dilation and length contraction in Appendix A.
Please forgive me for setting c = 1, but then later showing it explicitly, in an inconsistent manner, as well as my switching between a space-like and a time-like metric. Proper Time topology eliminates these confusions as the metric is fully positive, and all dimensions are naturally measured as distance.
|.||Special Relativity||Proper Time|
|Points:||(ct, x, y, z)||(x, y, z, s)|
|Local Metric:||-(cdt)2 + (dx)2 + (dy)2 + (dz)2||(dx)2 + (dy)2 + (dz)2 + (ds)2|
|Velocity:||(c, dx/dt, dy/dt, dz/dt)||(dx/dt, dy/dt, dz/dt, ds/dt)|
|Energy/Momentum:||m(c, dx/dt, dy/dt, dz/dt ) / √(1 - β2)||m(dx/dt, dy/dt, dz/dt, 1 ) / √(1 - β2)||Energy/Momentum:||m(cdt/ds, dx/ds, dy/ds, dz/ds )||m(dx/ds, dy/ds, dz/ds, dt/ds)|
Of course the conservation laws are unchanged. This is not a simplification for these laws, but the thing to note is that conservation laws are consequences of symmetries, and symmetries cannot be trusted. So the fact that Proper Time doesn't simplify the conservation laws is not significant in terms of its being an acceptable alternative. What's important is that the local topology, the place where the weird quantum physics takes place, is simplified.
The assumption that all frames of reference are equivalent, which is the fundamental symmetry law that underpins special relativity, is replaced in Proper Time with the assumption of the combination of a special topology and a constant speed for all particles. Instead of a theory based on a symmetry relation, Proper Time Topology is instead based on an assumption of simplicity. For example, since all electrons in Proper Time are travelling at the same speed, (at least locally), it is reasonable to assume that the waves that represent them have a characteristic frequency, one that does not depend on the velocity of the electron.
Intuitively, Proper Time has some simple explanations for some of the oddities of our universe. It's clear why we can't accelerate objects to faster than the speed of light, as that is the speed they are always travelling at. The huge amount of energy present in even stationary objects (i.e. E = m c2) is explained by their velocity in the hidden dimension. The distinction between massive and massless particles disappears, (except in the conservation laws).
Note that in special relativity, a particle's position, as a function of the global time, can be defined with just 3 numbers, but in Proper Time a fourth number is required. This means that when converting from special relativity to Proper Time, there is an undetermined variable. But since the error in this variable cannot be more than L/2, the effect of this undetermined variable will be negligible in practice.
Let W(x, y, z, s, t) be a plane wave function that represents a wave moving in the (kx, ky, kz, ks) direction. In order for the wave to travel at the velocity c, we need the frequency ω to be equal to k c, where k is the length of the k = (kx, ky, kz, ks) vector. A general form for W is W(k • x – ω t), where W is some cyclic function with period 2π. For a linear theory, we might try a cosine function, but in general the function is going to have to be highly non linear in order to get the right quantum statistics. I'll leave the interpretation of W for later. For now, suffice it to note that W is a general plane wave.
In order to satisfy the Proper Time topology, W must be single valued over s. This implies that L ks = 2nπ for n an integer. The wave vector is therefore quantized, at least in the proper time direction, according to:
(6.4) ks = 2 n π / L.
This implies that the speed in real space R3, as well as the time dilation ratio, Rtd = √(1 – (v/c)2) = ks / k is also quantized:
(6.5a) Rtd = ks / k
= 2 n c π / Lω,
(6.5b) v = c √ (1 - (2ncπ / L ω)2).
This result is very heartening, because the classical nature of Schroedinger's wave equation (see section 5.b) implies a quantization of velocity, and some sort of hidden length scale.
There are some other odd consequences of this quantization. The number of different possible speeds Ns is no longer infinite as in special relativity, but instead is finite (for a particle with a given characteristic frequency ω):
(6.6) Ns = [ω L / (2 c π)].
Where "" denotes greatest integer less than or equal. Of course for this theory to be realistic, Ns must be very large, so in the remainder of this note, it is so assumed.
Since there are only a finite number of different speeds, there must be a maximum speed, and this maximum is attainable, but is less than c:
(6.7) Vmax = c √(1 - (2cπ/Lω)2) ~= c - 2c(cπ/Lω)2.
There is also a minimum attainable speed, which may be zero, or may be larger, depending on the exact values of ω, c, and L. But the minimum attainable speed will be bounded above by:
(6.8) vmin < c √( 4cπ/Lω).
The above equation (6.8) gives the quantization of speeds for speeds near zero. That is, low speeds are quantized approximately according to:
(6.9) vslow = nc √( 4cπ/Lω) + vmin.
The quantization in velocity has a consequence in the equations of motion for very slow moving waves. The continuity relation that enforces the quantization of velocity has to be applied at a rate of once per each revolution that the particle makes in the S1 dimension. This implies that the slowest possible acceleration that a wave can be given is defined by a force that changes the velocity of the wave by one velocity step (i.e. equation 6.8) per revolution. Since the particle is assumed to be travelling at close to c in the s dimension, the result is that this characteristic minimum acceleration is given by:
(6.10) amin = c2 √( 4cπ/Lω) / L = √( 4c5π/L3ω).
Since it is impossible to achieve an exact velocity, it is also not possible to define the kinetic energy exactly. From this we can conclude that the wave/particle travels so as to either cause its kinetic energy to become more than the potential energy would indicate, or so that its kinetic energy becomes less. But since there is a minimum velocity, and this velocity is unlikely to be zero, it would make sense that the particle would tend towards choosing the larger velocity, and therefore the larger kinetic energy. The result is that very small forces will drive larger than expected accelerations. That is, the wave will tend to accelerate more than predicted by special relativity.
In order for this topology's effects to have gone unnoticed, it's clear that L must be almost vanishingly small. This requires ω to be very large in order to avoid the quantization effects of equations (6.4-6.10).
The physics literature does have mention of a minimum acceleration effect known as "Modified Newtonian Dyanmics" or MOND, but it appears to be such a large effect that it is not consistent with equation (6.10), at least when the Proper Time topology is used to explain the interference terms in quantum statistics. MOND was developed as an alternative explanation for the rotational curves of galaxies, which most astronomers ascribe to dark matter. The original paper is Mordehai Milgrom “A modification of the Newtonian dyanamics as a possible alternative to the hidden mass hypothesis”, Astrophysics. J. 270:365-370 (1983), and a search of the internet for the terms "MOND" and "Milgrom" will provide plenty of hits. The characteristic acceleration in MOND is about 1x10-10m/sec2. For more information on this fascinating topic, here are some weblinks: http://nedwww.ipac.caltech.edu/level5/Sept01/Milgrom2/Milgrom_contents.html http://www.astro.umd.edu/~ssm/mond/litsub.html
If the Proper Time topology is a real part of space time, then it will have to support Schroedinger's wave equation, and the interference effects (see equation 5.9) that make quantum statistics so different from the usual interference effects. Since interference effects are so much lower in frequency than the driving frequencies, we can expect that the interaction interference will be more easily observable.
Accordingly, let W1 and W2 be two interacting waves in the Proper Time topology, and let's compute their interference patterns. If the two waves both represent the same particle type, for instance electrons, then their ω values will be the same. This is due to the fact that all particles in the Proper Time topology travel at the same speed. But the two particles can have different wave vectors, so let their corresponding wave vectors be k1 and k2. Assuming equal magnitudes for the waves represented as cosines (and anything else is problematical due to the fact that this is a very non linear theory that we don't know the details of, but the average and beat frequencies, which is all we care about at this time, should be correct), then the interference is:
(6.11) W1(k1 • x
– ω t) +
W2(k2 • x
– ω t)
~= cos(k1 • x – ω t) + cos(k2 • x – ω t)
= cos((k1+k2)/2 • x - ωt) cos((k1-k2)/2 • x).
The (k1+k2)/2 term is of the same form as the original W waves, while the (k1-k2)/2 term is an interference term. Since the wave functions correspond to particles with velocity c, we can convert k1 and k2 to velocity format using the relation v = c k/k = c2k/ω. The resulting interference term is:
(6.12) cos((v1 - v2) ω/2c2 • x).
The above term gives the inteference between two plane waves. For more general waves, the above is integrated along a suitable path, say by dη, and the general interference term is therefore:
(6.13) cos(∫12 (v1 - v2) ω/2c2 • dη).
Comparing term (6.13) to equation (5.9) shows that interference between waves in Proper Time topology will be equivalent to interference between waves in Schroedinger's equation providing:
(6.14) ω/2c2 = m/
Consistent with the Proper Time topology, the above equation corresponds to the frequency of a particle travelling at speed c, but there is now an extra factor of two. This factor comes from the conversion between quantum and classical interference. Quantum interference is defined as a straight subtraction, but classical interference takes the difference and divides by that extra factor of two.
At the present time, the ω frequency is too high to be detected, hence the fact that a plane wave in quantum mechanics appears to have no spatial or time dependence, but the beat frequencies are detectable, and are the interference effects seen in Schroedinger's wave equation.
While it is a true that the the real extension of Schroedinger's equation supports arbitrary spin wave states, the ones generated in the most obvious manner (i.e. take equation 5.8 for spin-1, and replace φ with some multiple a of φ, the new equation will have spin-a), the resulting wave state is not stable. As ttime goes on, the dissipative term in equation (5.4) will cause the state to evolve to a standard solution of Schroedinger's equation. But the Proper Time topology has an extra dimension to play with, so it seems likely that stable solutions with other than the usual integer angular momentum exist. While the way to go about finding these is with separation of variables, an obvious spin-1/2 solution exists, and examining it will promote some familiarity with the equations.
For reference, here are the extended real Schroedinger's equations, where I've divided out the mass:
+ grad(v2 / 2) + grad V/m
h2 / 2m2)
grad(Δ (A) / A),
(7.1b) ∂A2/∂t + div(A2 v) = 0.
Schroedinger's real equation in Proper Time topology is identical, but in four dimensions instead of three, and the fourth dimension s is to be cyclic. The above equations show time evolution. For a stable solution, set the partial deriviatives with respect to t to zero, giving:
(7.2a) grad(v2 / 2) + grad V/m
h2 / 2m2)
grad(Δ (A) / A),
(7.2b) div(A2 v) = 0.
Now let's assume that our solution is to describe a single particle moving with constant velocity. We want to use all the ammunition available from the previous attack on the 3-dimensional Schroedinger's wave equation, so we need to separate variables into a 3-dimensional part and the s part.
When looking for a spin 1/2 solution, it's natural to consider setting up a solution where the magnitude of v matches that of a known 3-dimensional solutions, but where the average of v over the s dimension is half that of the 3-dimensional solution. Putting the s component of v to a constant, as is suitable for a particle moving at a constant velocity, makes the divergence in equation (7.2b) give the same result as for the 3-dimensional case. And with the magnitude of v the same as the 3-dimensional case, it only remains to set the A to be equal to the 3-dimensional solution to get all of equation (7.2a) to be the same.
In order to do this, we need to have a vector of unit length, whose direction depends on s, and whose average is 1/2. For a vector in the z direction, an obvious candidate is:
(7.3) (cos2(s), sin(s), sin(s) cos(s), 0)
The above vector has magnitude 1, but averages over s to (1/2, 0, 0, 0). In addition, it is cyclic in s, is continuous, and can be continuously rotated around the z-axis to provide a set of vectors with magnitude 1 that average to (cos(φ), sin(φ), 0, 0)/2. Note that those solutions to Schroedinger's wave equation in 3-dimensions which are eigenstates of the z-component of angular momentum, will only have velocities in the x and y directions, so it is clear that the above vector, along with its rotations around the z-axis, provide a continuous set of vectors with the appropriate properties to convert a 3-dimensional solution to Schroedinger's wave equation to a Proper Time solution.
The z-component of the angular momentum is computed as an integral over all space. Integrating out the s dependence (which only shows up in the vectors defined in (7.3a), leaves an average vector in the same direction as the standard 3-dimensional spin-1 solution, but with half the magnitude. It is therefore a spin-1/2 solution to the extended Schroedinger's wave equation in the Proper Time topology.
Relativistic quantum mechanics uses spinors to represent spin 1/2 particles, and these spinors have two components, with the extra component corresponding to anti-particles. While I have not shown that other solutions do not exist, nor that the transformation properties are the same (though the fact that the metric used in the Proper Time topology is identical to the standard one may convince some), it is tempting to assume that the anti-particle appears in the Proper Time topology as the same as the above, but with β defined to be negative.
As I work out more general solutions for Schroedinger's equation in the Proper Time topology, I'll add them here. But what we already have is a good start. Where QED requires different spaces to model scalar and spin-1/2 particles, the Proper Time topology gets them both as different solutions to the same equations.
This research began as an attempt to modify QED (Quantum Electrodynamics) so that a single, not terribly complicated, field would allow the modeling of more than just a single electron. It soon became apparent that the problem of quantum interference was at the heart at the difficulty in doing this. An analysis of quantum interference from the classical point of view, as illustrated by Bohmian mechanics, suggested that hidden dimensions needed to be included. Bohmian mechanics also provided an interpretation of the relationship between the extensions of Schroedinger's equation and Schroedinger's equation itself, and this suggested that the extensions should be used. An analysis of the results of quantum entanglement experiments, along with the fact of wave function collapse, suggested that a new interpretation of time would be required, and that new interpretation would have to be centered around the concept of proper time.
The equations of special relativity suggested that proper time should be treated identical to the 3 real dimensions, so the topology of Proper Time fell out naturally. When relativistic dynamics turned out to be simplified in Proper Time, as opposed to Minkowski space, the selection of Proper Time as the topology was shown to be likely to be correct. Calculations for self interactions showed no real problems with interpretation, given the very high energy/small distance nature of the proposed hidden dimension, and the interpretation of the interaction between multiple particles showed an interference effect that is identical to that of Schroedinger's equation.
The Schroedinger's wave equation is trivially generalized to the Proper Time topology,
and solutions to the regular Schroedinger's wave equation are still valid. In addition,
a solution with Lz = 1/2
h has been shown to exist.
This is all very heartening, (which is why I'm publishing this now), but there is still a lot of work left to do.
The first thing to do is to extend equation (5.4), which defines the dynamics of the velocity field, to the proper time topology. Since the conversion from the Proper Time topology to Schroedinger's equation involves the replacement of ω with m, it seems likely that m will be proportional to the amount of wave activity in the s dimension. That is, since those things travelling at the speed of light do not experience the passage of proper time and cannot have a non zero wave vector in the s direction, their waves will not depend on that dimension. Waves that do have a non zero wave vector in the s direction will presumably have mass, and that mass may be extracted in Schroedinger's wave equation as an average over a very small region of a term that might look, for example, like ∂2/∂s2. In any case, since the theory assumes that waves travel at the same speed in all directions, including s, it seems not unlikely that the full wave equation will involve all four dimensions in the same way. If this is the case, it may be possible to make some deductions as to the form of that equation. Then take the extended equation and see if we can get intrinsic spin to work on it. Since Schroedinger's equation cannot support spin 1/2, (but instead adds it axiomatically) it may be that intrinsic spin is a local feature of the topology that is averaged out when the s dimension is averaged over. Then the resulting theory needs to be shown that it, like classical mechanics on the Proper Time topology, is also compatible with special relativity.
One of the problems associated with extending quantum mechanics to extra dimensions is that one ends up with more particles, generally at the higher energies, than one desires. An example of this kind of issue will be put in this paper as Appendix B. But if the objective is to create a fundamental theory, then it is necessary that these higher energy solutions correspond to real particles observed in the world. The natural place to look for higher energy solutions would be the the higher energy correspondents to the electron. That is, given a field equation that supports solutions that represent electrons, it would be best if the same field equation also supports higher energy solutions that represent the μ and τ as well. Ideally, an accurate field equation should give the relative (bare) masses of these particles.
The relationship between the Kahluza-Klein extension of special relativity to include electromagnetism, and the Proper Time topology needs to be explored. Maybe this is an indication that more hidden dimensions need to be added to the Proper Time topology, or maybe the one that is already there is equivalent to that posited in Kahluza-Klein. The Proper Time topology is symmetric with respect to parity (i.e. space inversion), but it was shown in section [5.c] that the Proper Time topology would nevertheless support fermion wave functions. Time reversal, on the other hand, will be explicitly broken by any theory that connects wave functions, through time, to their collapsed predecessors. The operation of charge conjugation awaits the inclusion of electromagnetism to the Proper Time topology.
Rather than risk associating the teachers and institutions of higher learning that have had an influence on me with what may be treated as another crack-pot theory, I will here only acknowledge that the physics community has treated me and this theory with nothing other than respect and leave it at that. Carl Brannen
For those who have difficulty seeing how it is that the Proper Time topology, which explicitly possesses a preferred frame of reference, could be equivalent to special relativity, which makes the assumption that no such frame exists, this appendix provides detailed calculations for time dilation and length contraction both in special relativity and in Proper Time topology.
Special Relativity Solution: The voyage requires 3/0.6 = 5 years each way for a total of 10 years. This is the proper time experienced on the Earth. The spaceship experiences a time dilation of √(1 – 0.62) = 0.8, so the proper time experienced on the spaceship is 10 x 0.8 = 8 years.
Proper Time Solution: The spaceship starts at the point (x, y, z, s) = (0, 0, 0, 0). Align the x axis with the direction of travel. The velocity of the spaceship on the outgoing voyage is therefore given by the vector (0.6, 0, 0, 0.8). The 0.8 value is required to make the speed of the spaceship work out in total be 1. The spaceship’s position as a function of the global time t is therefore:
(0, 0, 0, 0) + (0.6, 0, 0, 0.8) t1
Setting this equal to (3, 0, 0, s1) gives t1, the global coordinate time for the arrival of the spaceship at its destination, and t1 is therefore 5 years. Note that the value of s1 is unspecified, as the total length of the hidden dimension is negligible as compared to the many light years of travel. Since the proper time component of the velocity of the spaceship is 0.8, the total elapsed proper time on the outgoing voyage of the spaceship is therefore 0.8 x 5 = 4 years. Similarly, the return trip uses a velocity of (-0.6, 0, 0, 0.8) and results in a coordinate time passage of 5 years and a proper time for the spaceship of another 4 years. The result is, of course, identical to the Special Relativity result.
Special Relativity Solution: The Lorentz contraction factor is 1/√(1 – (12/13)2) = 13/5, so the proper length of the rod is 6m x 13/5 = 15.6 meters.
Since the Proper Time topology does have a preferred coordinate system, the question is not as clear as it is in special relativity. But in any given coordinate system, the constancy of the speed of light provides a technique for measuring length. Accordingly, the rod can be measured in its own frame of reference by calculating the time required for light to travel the length of the rod. Since proper time is a property of individual particles, rather than dimensional objects such as rods, the length of the rod will have to be measured by computing the time required for the light to travel down the rod, be reflected at the end, and then travel back to the point of origin on the rod. The proper time experienced by the end point of the rod during this flight will indicate (when multiplied by c = 1) twice the length of the rod.
So let the rod begin at position (0, 0, 0, 0) through (6m, 0, 0, 0), and set the velocity vector for the rod to be (12/13, 0, 0, 5/13) so that it moves in the +x direction. The light signal starts at (0, 0, 0, 0) and proceeds with a velocity vector of (1, 0, 0, 0) until it meets with the other end of the bar at time t1. The light direction is then reversed, and it travels with velocity (-1, 0, 0, 0) until it meets up with the trailing end of the bar at time t2. The length of the bar, in the reference frame of the bar, is then 1/2 the proper time experienced by the trailing end of the bar from 0 to t2. The equations for t1 and t2 are therefore:
(0, 0, 0, 0) + (1, 0, 0, 0) t1 == (6, 0, 0, 0) + (12/13, 0, 0, 5/13)
(1, 0, 0, 0) t1 + (-1, 0, 0, 0)( t2 - t1) == (0, 0, 0, 0) + (12/13, 0, 0, 5/13) t2.
Since our real world does not distinguish between the hidden “proper time” coordinate, the equalities need only be established for the first three coordinates.
The solution is t1 = 13 x 6 meters, and t2 = 2028/25 meters. The proper time experienced on the trailing edge of the rod is, by time dilation, 5/13 of t2, which gives 156/5. Half of this is the proper length of the bar, which is the same as the value given by special relativity. Therefore, both theories show the Lorentz contraction of the bar to be the same.
This modification of Schroedinger's equation removes the imaginary numbers using a hidden dimension. Note that this is not the hidden dimension of the Proper Time topology, but this appendix is intended to illustrate what kind of effects hidden dimensions have on Schroedinger's equation. For reference, here's Schroedinger's wave equation:
h ∂Ψ / ∂ t =
(- h / 2m Δ + V(r)) Ψ
When we add an extra dimension to the above equation, the old functions Ψ will have to be extended to new functions Ψ' which instead of being functions of time and 3 dimensions, will now have to be functions of time and 4 dimensions. Call the coordinate for the new dimension ζ, and suppose that it is cyclic with length 2π. That is, Ψ'(x, ζ, t) = Ψ'(x, ζ + 2π, t). The simplest guess for how Ψ' depends on ζ is a sine or cosine.
The wonderful thing about sin(ζ) and cos(ζ) is that they convert into one another when differentiated with respect to ζ, but that going one way there is an extra minus sign. This is identical to how imaginary numbers work, if one interprets 1 as sin, i as cos, and replaces "multiplication by i" with a derivative with respect to ζ. Here's a table showing the equivalence.
|Values:||z = x + iy||x sin(ζ) + y cos(ζ)|
"Values" in the above table are the values of the Ψ and Ψ' functions, while "Action" refer to the operators in their respective equations, that is, in Schroedinger's wave equation and in this extension of Schroedinger's wave equation. The above table shows how to convert Schroedinger's wave equation from a complex equation in three dimensions (plus time), into a real equation in four dimensions plus time. The converted equation is as follows, note that the operator i has been replaced with a derivative with respect to ζ:
h ∂2Ψ / ∂ζ ∂ t =
(- h / 2m Δ + V(r)) Ψ
Any (complex valued) solution of Schroedinger's wave equation (B.1) can be converted into a (real valued) solution of the extended equation (B.2). The conversion, according to the above table, is as follows:
(B.3) Ψ'(x, ζ, t) = Re(Ψ(x, t)) sin(ζ) + Im(Ψ(x, t)) cos(ζ).
While every solution of Schroedinger's equation gives a solution to the extended equation, the reverse is not the case. That is, if Ψ' has ζ dependency other than just sine or cosine, the table above does not give a translation into a solution of Schroedinger's wave equation, at least the original Schroedinger's equation.
Instead, the extra solutions, when translated back from ζ into complex functions, give solutions to altered versions of Schroedinger's wave equation. The extra solutions correspond to wave equations for particles with masses or potentials different from the original solutions. It's also possible to consider translations of actions other than just i. For example, the action "multiplication by -1" can be obtained by two consecutive mutliplications by i, and is therefore translated as ∂2/∂ζ2. A more general table, this suggests that the above table be extended to show some of these more general translations:
|Values:||z = x + iy||x sin(n ζ) + y cos(n ζ)|
Another way of seeing the above translations is to use separation of variables on equation (B.2) in the usual way as is common in any textbook showing how to find general solutions of Schroedinger's equation.
Depending on how you choose to extend Schroedinger's wave equation, you can create equations that have, in addition to the usual solutions, solutions that correspond to particles with higher mass or different potentials (and therefore different charge). But while these are interesting, the solutions do not give the things that we'd like to have, such as the mass of the muon. For that, we need to have the non perturbational theory. But the fact that dimensional extensions of Schroedinger's wave equation creates equations with solutions for more than just one mass particle is heartening. This, along with the result that the Proper Time topology supports quantum interference effects, gives an outline for a multiparticle wave function that emulates the spin statistics used so successfully in quantum field theory.