Ideas about a Quantum Theory without Process Type 2

This article collects arguments for a quantum theory without process type 2. In quantum mechanics we have 2 different types of unitary processes. To cite Hugh Everett

Process 1: The discontinuous change brought about by the observation of a quantity with eigenstates  in which the state  will be changed to the state  with probability .

Process 2: The continuous, deterministic change of state of the (isolated) system with time according to a wave equation , where  is a linear operator.

Process type 1 is related to the state change of a conscious observer. Process type 2 shall describe an objective physical reality. The existence of these 2 process types gave us the measurement problem and the paradox of Wigner’s friend. Nobody currently knows why and when process type 1 takes place and when process type 2 evolves.

Decoherence Theory and Measurements

From decoherence theory we learn that we can gain knowledge about a “quantum system” only by destroying it. As long as there is no interaction with the quantum system, it can be described by process type 2 alone. However to find out whether this is really true, we must interact with the quantum system and observe some quantities. The observation starts with the coupling of our quantum system, i.e. the rest of the world, to the one that shall be observed. Technically this coupling means that the state vector of the measured system loses its existence as well as the state vector of the rest world does. In non-relativistic quantum mechanics both together must be described by the state vector of the whole world which lives in the product space of the spaces of the formerly independent systems.

An observable O that acts only on the measured system is represented by a linear operator , that acts only in a subspace of the product space. There is an infinite number of ways, how the world’s Hilbert space can be split into subspaces. With the selection of an observable we choose a certain split of the Hilbert space into 2 subspaces. This subjective choice brings the quantum system back to life for “observation”.

Technically the probabilities for measurement outcomes are given by the reduced density operator of the quantum system

 .

The degrees of freedom of the rest world are traced out completely. The expectation value of an observable O of the quantum system alone then is given by

 .

Decoherence theory tells us how the dynamical evolution according to process type 2 shapes the density operator and in which basis the entanglement of the subspaces is high. But still the measurement process has different outcomes that may enter consciousness with different probabilities. Decoherence theory does not solve the problem of outcomes as stated clearly for example in Schlosshauer’s book.

Because we destroy the quantum system when gaining information about it, it is generally not possible to gain complete knowledge about it. Only if the measurement (entanglement) basis fits to the possible states before the measurement, which depend on our preparation, we can gain complete knowledge.  Such a measurement is an ideal von Neumann measurement and establishes the von Neumann chain before it “collapses” according to process type 1 because of the observation.

For example according to Everett’s description of Wigner’s gedankenexperiment the observer W outside of the room thinks he has this chain with a superposition of his friend’s F state up to the end when W looks at the notebook.

But in F’s opinion process 1 took place earlier before W opened the door

As you can see probabilities  and outcomes are the same. However W has the possibility to find out whether F had been in a superposition by measuring in a basis  where

 

W’s and F’s chains then lead to different probabilities for the outcomes.

W gets this result meaning ”F had been in a superposition relating to the original basis” with probability 1, whereas F’s earlier process 1 leads to probabilities  for the different outcomes.

At least theoretically thus it should be possible to find out what is realized in nature.

The Many Worlds Interpretation

The many worlds interpretation tries to solve the paradox by denying the existence of process type 1 in favour for the continuous process type 2. The state of mind shall correspond to a state vector or density operator living in a subspace, just as for the states of matter or particles. The reason why F never experiences a superposition like

shall be that F has only access to a frog’s perspective where his state of mind is either the belief that there is a dead or there is a living cat. However from the bird’s perspective the superposition exists. But this way new problems are introduced:

While there may lie some truth in the radical suggestion to drop one of the process types, the many worlds interpretation is not only an interpretation of quantum mechanics. It is missing an explanation for the frog’s perspectives. It can often be read that a state vector shall split into its components (often or even continuously) thus creating the frogs’ perspectives. But the problem is, that a state vector does not have any components without choosing a basis. At any point of time it has components and has not, depending on the basis. Decoherence theory gives us pointer states and a basis, but why should these pointer states have something to do with a conscious perspective? Answering this question requires to modify quantum mechanics somehow. The answer is not included in current quantum theory.

There are no Big Jumps

The first figure above suggests that process type 1 allows big jumps while process type 2 requires some time to travel from A to B. In the figure the processes take place on a 2-dimensional surface of a 3-dimensional unit sphere. But in a space with many dimensions the difference between the process types may vanish.

In the smallest Hilbert space - a qubit - the distance between two unit vectors is restricted by

In a very large Hilbert space - of the world - the distance between any two unit vectors lies within the same interval [0,√2]. Imagine there are so many dimensions that you can always find one that gives you a shortcut from A to B. You might want to say that these 2 vectors, that are composed of qubits

have a bigger distance because several bits are different. But in contrast to the distance we defined first this is not an absolute measure. You can always find a basis where only 1 bit is different and then the same vectors may look like this

An infinite number of bases is totally equivalent to describe the same abstract Hilbert space vectors.

Thermodynamics and Black Holes

Consider these 2 equations from classical statistical physics

 

It defines the entropy S via purely classical mechanical quantities like energies E and H, particle numbers n, positions q and momenta p or more general canonical variables q and p.

However Planck’s constant appears in the denominator of the partition function . It enters the formula not before a comparison of the quantum mechanical entropy of 1 particle trapped in a box of a certain volume with its classical analogon. Planck’s constant puts a finite number of  dimensional areas into the classical phase space (Γ space). Their size is fixed but their shape is totally undefined.

The mechanical entropy definition is quite general (formula above is for 1 particle type, can be generalized for several types). For an ideal gas with particle energies independent of momenta  it is proportional to the volume V, i.e. the entropy is an extensive quantity depending on the amount of matter as already stated by empirical thermodynamics.

2 things are worth noting:

The entropy of black holes gives a partly similar, partly different picture. Nobody currently knows whether the Bekenstein-Hawking entropy gives us what is realized in nature, because an experimental test is not feasible. In the simplest case of a Schwarzschild black hole it gives us

 or  or

where M is the mass of the black hole, R the Schwarzschild radius of the horizon and A its surface area. Again a separated space - separated in thoughts, there is no wall at the horizon though some bring a firewall into the game - gets a finite entropy despite the fact that any position coordinate can encode an infinite amount of information. Again a non-absolute division of a continuum - here the spacetime - is achieved by considering quantum effects: the Compton wavelength of a particle falling into a black hole. Interestingly Bekensteins original work from 1973 mentions that if there wouldn’t be a quantum limit, the particle’s gravitational radius would be limiting. General relativity brought us the Planck length without the help of quantum mechanics.

In quantum mechanics a mixed state of the world would mean a new postulate. Where should it come from? However mixed states are popular in physics because they often deliver the correct results. For simplicity they mostly  are explained by “classical noise” entering a quantum experiment “somehow”.

In quantum theory any entropy must be explained by the von Neumann entropy of an entangled state, i.e. by

where  is the reduced density operator.

The Bekenstein-Hawking entropy is strange for several reasons that shall not be discussed here. But at this point we have the second strong hint that nature delivers us information in an absolute measure. And the process how information is delivered are the quantum processes described above.

No Continuum?

When functions of space or momentum or energy appear in quantum theory, they always mean components of abstract Hilbert space vectors in a chosen basis. They have no meaning a priori. For example you can think of space coordinates as indices of a very long n-tuple. Then a complex function of a space coordinate can be approximated by a long n-tuple

Physicists believe that the space-time continuum must break down somewhere at the Planck scale. The Planck length is around  and the Planck time . Imagine the indices would enumerate such very small Planck units. Then the size of the universe would limit the length of our n-tuples. Or the other way round: the lengths of n-tuples give us the size of our universe.

However in Hilbert space dimensions are not ordered per se. So the question is what orders indices in certain bases and therefore gives sense to statements like “this event happened near that one and soon after”? This additional structure is modelled as Hamiltonian operators in non-relativistic quantum theory. The matrix elements of an operator in a certain basis may strongly connect certain indices and weakly most of the other indices. So the operator may lead to an emergence of ordered index chains and thereby to pseudo continuous quantities like space, momentum and energy.

can be seen as the matrix representation of an operator in a basis, where it connects adjacent indices symmetrically (and cyclically) thus leading to a position-like variable.

The Role of Time

Clearly in relativistic physics space and time have no different quality and are transformed into each other by a change of the viewpoint or the metrics, for example by a Lorentz boost. Same applies for energy and momentum. All continuous quantities of relativistic physics could emerge as described above from an unordered fine graining with the help of ordering operators.

In quantum theory space, momentum and energy are only some of an infinite number of equivalent bases. Time and energy are paired by relativistic physics in the same way as space and time are. Therefore in relativistic quantum theory time must be a basis equivalent to all others. While time played a special role in classical mechanics, non-relativistic quantum theory and thermodynamics, it can’t do so any longer in relativistic physics, which includes relativistic quantum theory, which is the most accurate description of nature that we possess.

Very obviously time plays a special role in the world where human observers live. This is the reason why time has a special role in the older physical theories. The psychological time has a direction and the most famous equation with a direction of time is the phenomenological law of classical thermodynamics  or

 

There have been many attempts to derive this phenomenological law from more fundamental theories. However they all failed, and the reason is that all proven physical theories contain some type of T-symmetry, meaning that moving into the past is possible in the same way as moving into the future. Boltzmann’s H-theorem cannot be derived without an additional T-asymmetric postulate that is not part of statistical mechanics. A very exact investigation about the physical basis of the direction of time can be read in Dieter Zeh’s book.

The contradiction between directionless physical time (similar to space) and directed psychological time must be taken seriously. While there are attempts to make the direction of time physical, a physicist normally will feel uncomfortable with the idea of breaking space-time or energy-time relations that way. But where is the way out of the contradiction?

The way out could be keeping both, i.e. a physical time related to space and energy and therefore having no direction, and a directed psychological time related to consciousness.

Information and Meaning

The information content of a message  is related to the number  of distinguishable modifications it can cause in a receiver

At first it depends on the message and the receiver.

Physically a message is a state vector in some subspace. A qubit, the most simple Hilbert space, can deliver a maximum of 1 classical bit of information to an observer after a “measurement”. A continuous Hilbert space can deliver an infinite amount of classical information, for example about the [continuous] position (x,y,z) of a “particle”. If there shall be an absolute measure of information in nature as implied by the thermodynamic and Bekenstein-Hawking entropies, there can’t be a continuum. If there were a continuum in nature, any finite measure of information would require a cultural agreement between sender and receiver on how to separate the continuous message into a finite number of parts. But without a continuum information gets the chance to become a purely physical concept.

Meaning is the concrete change in a receiver that is caused by a message. There is no meaning in any message, because the receiver is not part of the message by definition. In quantum theory the message therefore is a message only as long as it is not entangled with the receiver. With the entanglement the message loses its existence.

There is no meaning in any information. There is no meaning in this text. There is no text here. Rather you only see some dark and bright pixels on a computer screen. But there are no pixels at all. Rather there are crystals emitting light in different ways. But there are no crystals. Rather there are electromagnetic and lepton fields. But there are no fields, … and so on.

All our physical theories contain and relate mathematical symbols. They work if used by people who “understand” them. Without the people they mean nothing. Already in classical mechanics space and momentum can be exchanged by canonical transformations. The descriptions of nature in many pictures are mathematically equivalent. But which is the “right” picture?

With the additional relations between indices that come from the introduction of operators in Hilbert space pseudo continuous variables may emerge. But which of these variables is an energy and which a particle position? This question cannot be answered without the receiver. Energies and particle positions appear as such not before state changes of a conscious observer that is coupled to the Hilbert space.

Only Process 1

Is there something special of the jump destination of process type 1 which is related to consciousness? First of all it is a Hilbert space vector like all others. So why jump there and not anywhere else, for example onto a state

 

The difference arises when we cut out a subspace of the Hilbert space. Suddenly there are world vectors that look like a mixed state (a density operator) in the subspace, and few others that are pure in the subspace, i.e. they are a product of a vector in the subspace and another vector in the rest of the world. The jump destination is always a pure state. But why should we separate the Hilbert space exactly in that way and not in any other of the infinite many ways? The reason may be, that consciousness includes a certain separation. Hilbert space separation is a property of a conscious entity.

What is really strange in quantum theory is the existence of 2 process types and the unanswered question of why and when they alternate. While the many worlds (or many minds) interpretation denies the existence of type 1, we try to get rid of type 2. This means the continuous evolution of the world state vector actually shall be a sequence of fine jumps. Thus we get a non-linear stochastical collapse theory. Bassi and Ghirardi state, that non-linearity implies randomness and vice versa. We already discussed that there are only short distances in a high dimensional space. A Schrödinger equation could be seen as a differential equation of macroscopic variables like the heat equation . The apparent smooth motion of the probability amplitude could be the result of a stochastic process of fine jumps. The overwhelming amount of dimensions would not reflected by the simple macroscopic equation.

But now we are facing a new old problem. Why are there 2 alternating process types, i.e. a process type 1 related to a conscious entity and a process type 1 unrelated to any consciousness? We can choose a similar way out as in the many minds interpretation: There are no unconscious jumps, every change of the world’s state enters some conscious entity.

This last postulate might look monstrous at a first glance. But in fact it is only the extrapolation of an epistemology on a path which has been directed by modern science. The main reason why we did not recognize other conscious entities in the past has always been the lack of communication channels. Science gave us more channels, and at least apes, dogs and perhaps worms, plants and forests must be regarded as conscious entities. Of course it is still a long journey to a world of myriads of conscious entities wrapping subspaces of qubits.

Interestingly psychologists have theories of perception that resemble the consequences we are faced to after getting rid of process type 2 in physics:

What is new with this idea?

In 2003 Bassi and Ghirardi gave an overview of attempts to solve the measurement problem:

Some aspects of this idea already appear in other attempts, but this idea cannot be filed into the figure because it claims:

Normally the state of the brain is supposed to be or at least largely corresponds to the state of a mind, of one mind. And the state of a second brain should correspond to the state of a second mind. Therefore in quantum models that state of mind is often shown as [reduced] density operator or even as state vector.

While a density operator might be an adequate description of a brain, it is not so for a mind. There is no state of mind living in a subspace. There is only 1 state vector, and this 1 state vector is the state of all brains together.

There are many different views onto the common state. Perception occurs along a split of the whole configuration space and is strongly related to changes of entanglement, that can be experienced on this conscious plane. From the infinite number of bases for the whole space the split emphasizes such, that can be built from products of subspace base vectors.

The world state vector together with a conscious plane separating the Hilbert space are sufficient to uniquely define a Schmidt decomposition of the state vector

The unique  deliver the classical probabilities for the outcomes that are chosen by accident - or by the conscious plane enriched with some  representing its will.

A brain is the result of mind activity and not the other way round. Today’s quantum physics is merely describing the communication channel between minds.

Prime Numbers Forbidden

The requirement that a Hilbert space shall contain conscious planes excludes prime numbers from the set of possible dimensions for Hilbert spaces. The space must be separable into at least 2 parts. The dimension of each part must be > 1, because the product space of a Hilbert space with a one dimensional space would be itself. Thus the smallest possible space with conscious planes is four-dimensional. Possible dimensions are {4,6,8,9,10,...}.

“Big” and “Small” Planes

A plane that separates the space into a qubit and a bigger rest will in general perceive only few changes of the Schmidt decomposition caused by jumps of the state vector. The dimension of the smaller part is therefore a measure for the possible richness of perceptions, and of course it determines the maximum possible entanglement entropy at the same time.

If there should be a will that plays a role in jumping from an entangled state (as observed from the perspective of a conscious plane) to one of the offered outcomes, the dimension of the smaller part would be a measure for its might.

Chances and Challenges

Concerning the mind body problem most scientists still seem to hope that someday science can explain how the mind is generated by a configuration of matter. Recognizing consciousness as primary ontic entity saves us from this task. Putting it everywhere saves us from the need to explain how it gets localized in position space. There is the chance to connect physics to psychology, where models for will, pain and joy exist already, thus ending the dualism of mind and matter. Philosophers always told us that you may deny the existence of matter, but you can’t deny the existence of a mind that experiences its own being (“cogito ergo sum”).

Introducing “hidden variables” for the psychological times could help to understand why the second law of thermodynamics is working in practice. The jumps end on a product state (for those conscious entities that perceive the jump) with an entanglement entropy of 0. Starting from there entanglement can only grow like thermodynamic entropy.

The big challenge is the development of a microscopic theory which reproduces current quantum theory as macroscopic theory. The idea is not new and has first been picked up by von Weizsäcker et al. and called Ur theory.

Omitting process type 2 has also experimental consequences. Whenever there are several observers observing the same von Neumann chains, they must jump together [simultaneous in the non-relativistic limit]. A measurement of “is he in a superposition?” in an appropriate measurement basis must always lead to the outcome “no”.

Emergence of Time

Time evolution in non-relativistic quantum mechanics is boringly simple. A closed world has a Hamiltonian that is not explicitly time dependent. Therefore any whole-world Schrödinger equation can be integrated at once.

On the right hand side the Planck constant has been absorbed into energy and time so that only dimensionless parameters are left. The common non-relativistic time t can now be interpreted as emerging from a big number n of equal unitary jumps. The unitary jumps in turn could have jitter, i.e. not each  is equal, rather

Since

with the choice of  as elementary unit of time we arrive at

The smallest time that can be measured in 2018 is , while the Planck time is . This would mean the smallest change we can observe is the result of  jumps of the state vector.

Is there a chance to observe the difference from Schrödinger dynamics?

We set

 

where E shall be any eigenvalue of the Hamiltonian. The more x differs from 0, the bigger the difference between the exponential function  and the approximation  is.

The relative error

quickly increases when leaving the origin.

However energy eigenvalues of quantum systems are small. Even 1 TeV means .

The very accurate electron g-factor measurement has a relative error of around . However n is very big, not only 10000 as in the diagrams. So there is no chance to detect a deviation, if the  are chosen appropriately. But of course nature needs not to choose them in a way to reproduce Schrödinger dynamics as accurate as possible.

2 Qubits Space

A space of 2 qubits is the smallest space with vectors that can appear entangled. Consider 2 bases in this space. Let us call the first one the 10 basis. Its base vectors we name . The second base we call the +- basis with base vectors . The unitary transformation that transforms us one base into the other shall be U.

Base vectors of the +- basis shall be entangled in the 10 view and vice versa. For example these 2 vectors

are entangled in the 10 view and not entangled in the +- view.

We want a rotation of the state vector approximated by jumps. There shall be 2 conscious planes in the 2 qubit space corresponding to the 2 views. I.e. the 10 plane shall separate the whole space into 2 parts labeled (1) and (2), so that , and the +- plane shall separate the space into 2 different parts labeled (3) and (4), so that .

Assume the initial state vector is .  The 10 plane sees an unentangled vector. To reproduce the Born rule the probability to change the state into  must be 1 and the probability for the outcome  must be 0. The 10 plane thus will not change this state vector. But since the Schmidt decomposition is

in the +- view, the +- plane sees entanglement. With equal probabilities 0.5 one of the outcomes  and  is selected when this plane perceives. After perception the situation is inverted: the +- plane will not change the state while perception in the 10 plane will change it. The 2 conscious planes drive the state vector forever. However the movement will resemble more a Brownian motion than a rotation, because the probabilities for “forward” and “backward” jumps are equal. In this example the jump destination is always a base vector of the bases related to the conscious planes. Of course in general this must not be the case.

A rotation-like movement requires an internal state somewhere, a “hidden variable”, so that when the initial state has accidentally started a clockwise movement, the probability for further clockwise movement is higher and the probability for counter-clockwise movement is lower. This could be achieved for example by postulating:

After perception of a new [unentangled] state (after the jump), the probability for this jump destination is reduced by a hidden variable in the conscious plane that has triggered the  jump until the next jump to be triggered by this plane.

Numerical Simulations

todo…

Relativistic Dynamics

todo, e.g. free Dirac equation: physical time like space, relation to psychological time?

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