Quantum Fields and Impulses

Understanding quantum physics requires a paradigm replacement. Forget the idea that particles are little planets controlled by forces from classical physics. Instead one must approach quantum physics from an informational paradigm point of view. Fundamental particles are actually impulses (delta functions) which indicate that a state involving its quantum field has changed. Forcing a decision about which state exists in a quantum field with a measurement produces an impulse.

A simple finite state machine used in digital devices. Its pattern of inputs defines what its output state will be after commanded to change by the leading edge of the clock pulse. (from https://computationstructures.org/notes/fsms/notes.html)

State Machines

(July 7 ,2022) Computers and all digital devices are finite state machines. They use a global clock signal to indicate when to change state. These states are binary represented by either a 0 or 1.

In contrast to computers, the brain and the universe have no global clock signal and their states are continuous. Their information processing is described by soft state automata which in the case of the universe is quantum physics.

The actual state changes in both is done by impulses which in finite state machines are either the leading or lagging edge of the clock pulse signal. In physics these are the fundamental point particles.

Don Lincoln of Fermilab near Chicago provides the shortest and simplest introduction to quantum fields. If this whole idea is new to you start here.

Quantum fields (Potential Probability Fields) are Fundamental

(July 7, 2022) Most people know quantum mechanics has something to do with energy type waves. These waves are actually moving quantum fields like light waves but are static fields when not moving like electro-magnetic fields.

Particles come out of quantum fields under certain conditions. Fundamental particles take up no space themselves and actually only represent events, that is, state changes in the field. Mathematically they are represented as impulses.

Particles as Signals from Quantum Fields by Arvin Ash

Quantum Fields by Arvin Ash

Quantum Fields by David Tong of the Royal Institution, Cambridge

Generation of an impulse.

Impulses Represent State Changes

(July 7, 2022) In binary systems states are represented by 1 or 0, that is, a high voltage or low voltage generated by a switch. Looked at over time this produces a step function as shown at the top of the figure. The actual change in voltage levels is represented by an impulse which is also called the Dirac Delta function. This is properly found by taking the directional (Dini) derivative of the step function.

In practice a global clock signal changes the state in all binary devices and that signal is a pulse (an up step then down step). The state actually changes during the leading or lagging edge of that pulse. This shows the impulse is the thing which actually changes state.

Present day mathematics is built upon continuous functions and not discontinuous functions such as the step function. This left the impulse without a good mathematical definition yet it was needed by quantum physics. Paul Dirac (Nobel prize winning) was forced to do some ad-hoc definitions which is why the impulse came to be called the Dirac Delta. He presented two definitions showing just how uncomfortable he was with his derivations. (Dirac 1930)

The first definition was to present the impulse as the integral of some arbitrary shaped area with the area set equal to one. First this arbitrary shape is shrunk along the x axis to its limit (becomes so narrow that any further shrinkage has no further effect) so that it becomes like a vertical line. Once that is done the mathematical operation of integration is used define the area which is then arbitrarily set equal to one. The problem with this is that it ends up using two limits in sequence which is a contradiction of the whole idea of a limit. Once something becomes so small that it is meaningless it cannot become meaningless again by shrinking it in a different way. In this case one limit was for the integral area and the other one was for the impulse shape. Because of this, Dirac went on to call impulses “improper” functions:

“(x) is not a function of x according to the usual mathematical definition of a function, which requires a function to have a definite value of each point in its domain, but is something more general which we may call an ‘improper function’ to show up its difference from a function defined by the usual definition. Thus (x) is not a quantity which can be generally used in mathematical analysis like an ordinary function, but its use must be confined to certain simple types of expression for which it is obvious that no inconsistency will arise.” (page 58 or the 4th edition)

Because of this rather inelegant approach he tried to derive the impulse again using a method called “integration by parts” which is only valid for continuous and differentiable functions as that technique originates out of the differential chain rule of calculus. Because the impulse is not a continuous function this second approach is also invalid.

Still, his approaches are the best continuous mathematics can do leaving one or the other of Dirac’s derivations to be repeated in textbooks to this day. Of course, by not using directional derivatives this cripples the formulation of physical theories.

References

Dirac, P. (1930) Principles of Quantum Mechanics, Fourth Edition (Oxford University Press, Oxford, 1958, first published in 1930)

Dice as used in role playing table top games. At least here the probabilities are known.

Potential Probability

(July 7, 2022) Imagine you are given one dice but you do not know how many sides it has. The number of sides may range from 4 to 8 with the average number of sides being 6 and to keep things easy, its side values range from 1 through 6 randomly distributed. We can say the probability is 1/6 but that number is very imprecise because we do not really know how many sides the die has. It is a potential probability instead of an actual probability.

Quantum waves are potential probabilities which only become probabilities after they have been constrained by their environment (the wave functions are squared). Their uncertainty is with space and time instead of with the number of dice sides. Quantum waves are "unreal" with an imaginary number in their equations so they are not limited by the speed of light. The waves can thus sniff out their contextual situation and define the probabilities of event productions (impulses, particles)

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