We can consider a quantum wave as an area of space where a particle might be found. Schrödinger thought that this idea was silly and that the natural state of particles *is* the wave form. So we can say a particle-like interaction could occur somewhere in an atom’s electron field, for instance. The bottom line is that a single particle can exist in field form, that is, its position can be distributed over some area of space.
I have argued elsewhere that in the particle’s own frame there would be no field and so the field exists as a phenomena between inertial frames just as other weird changes such as those predicted by Special relativity do not occur within inertial frames, for instance fast moving rockets do not get shorter nor do clocks run slower by the measure of the rocketeers.
Schrödinger’s equation shows us that not only spatial location but any properties a particle can have can be in superposition, that is, the wave can be distributed over various outcomes just like a particle’s field can be distributed over various locations in space.
If we scaled up this phenomena (superposition and the Schrödinger wave equation) to the classic Schrödinger’s Cat analogy we have the state of the cat spread across dead or alive until the lid is opened. For the classic wave we have the cat somewhere in the box and we may assign probabilities to various locations but the cat isn’t anywhere in particular until an interaction occurs, that is, until it is detected. We normally think of the cat’s activities as being unknown when the lid is closed, but to be truly analogous to the quantum world we should consider that the cat’s location, movement, speed, path etc within the box are known only at the time of the lid closing and later when it is open but that its position and motion are unknown in the between time. By thinking about a cat sitting in the box we only isolate the events and keep other variables constant which would not be the case in real particle interactions, for instance if it were electrons in the box they would have cardinal but no ordinal property at all (you can’t identify them individually or say how they get from one place to another).
Feynman had an interesting way of explaining this saying that if you shine a light onto a target then the average motion of the photons is from the torch to the target, as we’d expect, but when you think about each individual photon things are not so clear, a single photon can go anywhere and Feynman extends this by saying that a photon hitting the target could have travelled all over the universe on its way to the target, pointing to ‘the universe’ as he explains this :)
Apart from a spread of possible locations and a spread across various outcomes, a third kind of wave seems to have been ignored, or perhaps it is simply assumed, but it is an essential wave form if we are to understand the strange world we inhabit. This is what I would call a ‘Temporal Wave’.
Let’s assume an event occurs within some interval. That event is like a particle of time, it occurs at a particular location on a time line. A wave form of this event would see the event as occurring at any one of a number of temporal locations along the timeline. That is, the location of the event on the time line, just as the location of the cat in the box or the outcome of an event (superposition) is spread across a set of possible locations until it is detected (until it interacts with an observer or another system).
How would we model this in the cat analogy? Let’s say we have an event, the vile of poison is released by the chain of events starting with the random emission of a radioactive substance (Schrödinger’s model). The wave form would say that this event can occur at any point with in the interval the cat is in the isolated (unopened) box (it is ‘smeared’ along the timeline).
A fourth form of wave is both conventional but a little mind numbing. If the cat’s inertial frame is isolated then we can not tell the cat’s relative speed until the observer open the lid. This does not scale up quite so well because the cat in the box is imagined as being at rest in the observer’s inertial frame, but in particle physics an isolated particle really is isolated and so its motion is unknown. The frequency of the ticking of a clock placed in with the cat can be anything within the range of possibilities, from zero to infinite. We can reverse this by asking how fast your clock is ticking relative to another observer. An observer passing you at the speed of light would measure your clock as being frozen, an observer at the event horizon of a Black Hole would measure your clock frequency as infinite. If your frame is isolated then the wave of your clock speed can be spread across that entire range. We actually see this by analogy in the way that the same clock, say a clock inside a rocket, is measured as having a different rate depending on the motion of the observer.
So, altogether there are four forms of the wave: until the lid is opened
1) The cat can be anywhere within the box;
2) Variable properties of the cat can be in superposition, such as the cat being alive or dead;
3) Events, such as the cat dying, can have occurred anywhere within the interval in which the cat is isolated in the box;
4) The interval that the cat is in the box is spread across a wide range of possible intervals (the wave of the cat’s clock rate is spread across all possible rates).
In other words, just as the location of the cat (or particle) is ‘smeared’ over many locations in space, and just as the outcomes are ‘smeared’ over all the possible outcomes, events are ‘smeared’ over all possible temporal locations and the clock frequency is spread across all possible rates relative to the observer who is isolated from the cat’s inertial frame. If the cat is truly isolated, like a particle, then when we open the lit we don’t know where it has been or how long it has been away although just as we see the outcome (cat alive or dead) and the location of the cat in the box we would also see evidence of the interval and from that we can calculate relative clock rate, for instance if only a skeleton of the cat remains then we can conclude that years have passed and the clock was speeded up (or our clock slowed down relative to the cat’s Rolex).
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