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Christoph Schiller - Motion Mountain - Notes

Background space
Every observer introduces his own background. It does not need to coincide with physical space (and it does not do so at the location of matter or black holes). But every observer’s background is continuous and has three spatial and one temporal dimension. Continuous background space is introduced by the observer only to be able to describe observations. We do not state that background space and time exist a priori, as Immanuel Kant states, but only that background space and time are necessary for thinking and talking, as Aristotle states. In fact, physical space and time result from strands, and thus do not exist a priori; however, background space and time are required concepts for any description of observations, and thus necessary for thinking and talking.

Strands and fluctuations
Nature is made of unobservable, fluctuating, featureless strands. Strands are one-dimensional curves in three-dimensional space that reach the border of space. All strand fluctuations are possible, as long as strands do not interpenetrate, there is no speed limit for strands.

Fluctuations change the position, shape and length of strands; fluctuations thus change position, orientation and phase of strand crossings. Fluctuations randomly add detours to particle strands and randomly shift the core position. Fluctuations do not keep the strand length constant. Fluctuations do not conserve strand shape nor any other property of strands. However, fluctuations never allow one strand to pass through another. Due to the impenetrability of strands – which itself is a consequence of the embedding in a continuous background – any disturbance of the vacuum strands at one location propagates.

Physical systems are surrounded by a bath of fluctuating vacuum strands. The properties of fluctuations, such as their spectrum, their density etc., are fixed once and for all by the embedding.

Crossings and observable switches
A crossing of strands is a local minimum of strand distance. The appearance of a crossing does not depend on distance or on the number of strands in between.

A crossing switch is the rotation of the crossing orientation by an angle π at a specific position. More precisely, a crossing switch is the inversion of the orientation at a specific position. Strands are impenetrable; the switch of a crossing thus always requires the motion of strand segments around each other.

Strands fluctuate in a background space, and only crossing switches can be observed. Events are (one or several) observable crossing switches of unobservable strands. All observations, all change and all events are composed of the fundamental event, the crossing switch. Any event, any observation, any measurement and any interaction is composed of switches of crossings between two strand segments.

All measurements are electromagnetic. In other words, all measurements in nature are, in the end, detection of photons. And the strand model shows that photon absorption and detection are intimately related to the crossing switch.

Tangle function
Continuity of any kind – of space, fields, wave functions or time – results from the time averaging of crossing switches.

The position, orientation and phase of a crossing are defined by the space vector corresponding to the local minimum of distance.

The tangle function of a system described by a tangle is the short-time average of the positions and the orientations of its crossings. The tangle function at any given time is not observable, as its definition is not based on crossing switches, but only on crossings. However, since crossing switches only occur at places with crossings, the tangle function is a useful tool to calculate observables. In fact, the tangle function is just another name for what is usually called the wave function. In short, the tangle function, i.e., the oriented crossing density, will turn out to describe the quantum state of a system.

Using the tangle function, we define the strand crossing position density, or crossing density, for each point in space, by discarding the orientation information, counting the crossings in a volume, and taking the square root. The crossing density – more precisely, its square root – is a positive number, more precisely, a positive real function 𝑅(𝑥, 𝑡) of space and time. A tangle function also defines an average crossing orientation and a average phase at each point in space. The average crossing orientation and the average phase are related to the spin orientation and phase of the wave function. The mathematical descriptions of these quantities depend on the approximation used.

The absolute value 𝑅(𝑥, 𝑡), of the wave function at a point should be related to the vectorial sum of all inverse shortest crossing distances at that point.

The tangle core, the region where the strands are linked or knotted, defines position, speed, phase and spin of the particle. The position of a particle is given by the centre of the averaged tangle core.

Motion of any quantum particle is the change of the position and orientation of its tangle core. Free quantum particle motion is due to fluctuations of tangle tails. The deformations of the tangle core are not important for free motion. The particle position is thus the average of all its crossing positions.

The phase of a matter particle is given by half the angle that describes the orientation of the tangle core around the spin axis. The particle phase is thus the average of all its crossing phases. The spin orientation of a matter particle is given by the rotation axis of the core. The spin orientation is thus the average of all its crossing orientations. The wave function of a matter particle is a blurred rendering of the crossing of its fluctuating strands.

The conservation of energy and momentum implies that the time average of the tangle fluctuations also conserves these quantities.

Vacuum
Vacuum is a collection of fluctuating, unknotted and untangled strands. Vacuum, or physical space, is formed by the time average of many unknotted fluctuating strands, strands in vacuum are not tightly packed.

Physical space is a homogeneous distribution of crossing switches.

Flat vacuum has a tangle function that vanishes everywhere.

Particles
A quantum particle is a tangle of fluctuating strands. A tangle is a configuration of one or more strands that are linked or knotted. Strands without knots are sufficient to recover modern physics. Tangles are characterized by their topology, i.e., by the precise way that they are linked or knotted. In the strand model, elementary particles are (families of) tangles of strands.

Quantum particles are particles whose tails cannot be neglected. The tangle tails reach up to the border of space.

Spin
Rotating tangles model spin. Spin is core rotation.

An elementary matter particle, or fermion, is a tangle of two or more strands that realizes the “belt trick”. The “belt trick” is the observation that a belt buckle rotated by two full turns – in contrast to a buckle rotated by only one full turn – can be brought back into its original state without moving the buckle; only the motion of the belt is necessary. The belt trick is also called the scissor trick, the plate trick, the string trick, the Philippine wine dance or the Balinese candle dance. It is sometimes incorrectly attributed to Dirac. The various options of the belt trick are related to the difference between matter and antimatter and to the parity violation of the weak interaction.

For tangles made of one strand – thus with two tails to the border – a rotation of the tangle core by 2π restores the original state. Such a tangle, shown in Figure 29, thus behaves like a spin 1 particle.

All spin 1/2 particles are made of two (or more) tangled strands, and thus have four (or more) tails to the ‘border’. For such tangles, a rotation by 4π of the tangle core – thus a rotation by two full turns – can bring back the tangle to the original state, provided that the tails can fluctuate. An object or a tangle core that is attached by (three or more) tails to the border of space can rotate continuously. The intermediate twisting of the tails that appears after rotation by only 2π corresponds to a multiplication of the wave function by −1, again as expected from a spin 1/2 particle. Any system that returns to its original state after rotation by 4π is described by spin 1/2. A simple exchange of two spin 1/2 particles (tangles, cups on hands, belt buckles) is equivalent to a multiplication by −1, i.e., to twisted tangles, arms or belts. In contrast, a double exchange of two spin 1/2 particles can always be untwisted and is equivalent to no exchange at all.

Base units
Every Planck unit is an observer-invariant limit value. Defining observables with the help of crossing switches automatically makes the Planck units 𝑐, ℏ, 𝑐4/4𝐺, 𝑘 and all their combinations both observer invariant and limit values.


 * Planck’s quantum of action ℏ/2 appears as the action value associated to a crossing switch. The action ℏ corresponds to a double crossing switch, or full turn of one strand segment around another. The physical action of a physical system evolving from an initial to a final state is the number of crossing switches that can be measured. Action measurement is thus defined as counting crossing switches. Physical action is thus a measure for the change that a system undergoes.
 * The (corrected) Planck length 𝑙Pl = √4𝐺ℏ/𝑐3 appears as the effective diameter of strands. Since the Planck length is a limit that cannot be achieved by measurements, strands with such a diameter remain unobservable. The distance between two particles is the maximum number of crossing switches that can be measured between them. Length measurement is thus defined as counting Planck lengths.
 * The Planck entropy, i.e., the Boltzmann constant 𝑘, is the natural unit associated to the counting and statistics of crossings. The entropy of any physical system is related to the logarithm of the number of possible measurable crossing switches. Entropy measurement is thus defined through the counting of crossing switches. The strand model thus states that any large physical system – be it made of matter, radiation, empty space or horizons – has entropy.
 * The (corrected) Planck time 𝑡Pl = √4𝐺ℏ/𝑐5 appears as the shortest possible duration of a crossing switch. Crossing switches that are faster than the Planck time do not play a role, as they are unobservable and unmeasurable. The time interval between two events is the maximum number of crossing switches that can be measured between them. Time measurement is thus defined as counting Planck times.
 * Energy is action per time. Now, the Planck constant is the unit of action, and is defined by a crossing switch. A system that continuously produces a crossing switch for every Planck time running by thus has Planck energy. An example would be a tangle that is rotating extremely rapidly, once per Planck time, producing a crossing switch for every turn.
 * Momentum is action per length. A system that continuously produces a crossing switch whenever it advances by a Planck length has Planck momentum. An example would be a tangle configuration that lets a switch hop from one strand to the next under tight strand packing.
 * Force is action per length and time. A system that continuously produces a crossing switch for every Planck time that passes by and for every Planck length it advances through exerts a Planck force. A tangle with the structure of a screw that rotates and advances with sufficient speed would be an example.