# Relational quantum mechanics

This article is intended for those already familiar with quantum mechanics and its attendant interpretational difficulties. Readers who are new to the subject may first want to read the introduction to quantum mechanics.

Relational quantum mechanics (RQM) is an interpretation of quantum mechanics which treats the state of a quantum system as being observer-dependent, that is, the state is the relation between the observer and the system. This interpretation was first delineated by Carlo Rovelli in a 1994 preprint, and has since been expanded upon by a number of theorists. It is inspired by the key idea behind special relativity, that the details of an observation depend on the reference frame of the observer, and uses some ideas from Wheeler on quantum information.[1]

The physical content of the theory has not to do with objects themselves, but the relations between them. As Rovelli puts it:

"Quantum mechanics is a theory about the physical description of physical systems relative to other systems, and this is a complete description of the world".[2]

The essential idea behind RQM is that different observers may give different accounts of the same series of events: for example, to one observer at a given point in time, a system may be in a single, "collapsed" eigenstate, while to another observer at the same time, it may appear to be in a superposition of two or more states. Consequently, if quantum mechanics is to be a complete theory, RQM argues that the notion of "state" describes not the observed system itself, but the relationship, or correlation, between the system and its observer(s). The state vector of conventional quantum mechanics becomes a description of the correlation of some degrees of freedom in the observer, with respect to the observed system. However, it is held by RQM that this applies to all physical objects, whether or not they are conscious or macroscopic (all systems are quantum systems). Any "measurement event" is seen simply as an ordinary physical interaction, an establishment of the sort of correlation discussed above. The proponents of the relational interpretation argue that the approach clears up a number of traditional interpretational difficulties with quantum mechanics, while being simultaneously conceptually elegant and ontologically parsimonious.

## History and development

Relational quantum mechanics arose from a historical comparison of the quandaries posed by the interpretation of quantum mechanics with the situation after the Lorentz transformations were formulated but before special relativity. Rovelli felt that just as there was an "incorrect assumption" underlying the pre-relativistic interpretation of Lorentz's equations, which was corrected by Einstein's deriving them from Lorentz covariance and the constancy of the speed of light in all reference frames, so a similarly incorrect assumption underlies many attempts to make sense of the quantum formalism, which was responsible for many of the interpretational difficulties posed by the theory. This incorrect assumption, he said, was that of an observer-independent state of a system, and he laid out the foundations of this interpretation to try to overcome the difficulty.[3]

The idea has been expanded upon by Lee Smolin[4] and Louis Crane,[5] who have both applied the concept to quantum cosmology, and the interpretation has been applied to the EPR paradox, revealing not only a peaceful co-existence between quantum mechanics and special relativity, but a formal indication of a completely local character to reality.[6][7]

David Mermin has contributed to the relational approach in his "Ithaca interpretation."[8] He uses the slogan "correlations without correlata", meaning that "correlations have physical reality; that which they correlate does not", so "correlations are the only fundamental and objective properties of the world". The moniker "zero worlds"[9] has been popularized by Ron Garret[10] to contrast with the many worlds interpretation.

## The problem of the observer and the observed

This problem was initially discussed in detail in Everett's thesis, The Theory of the Universal Wavefunction. Consider observer ${\displaystyle O}$, measuring the state of the quantum system ${\displaystyle S}$. We assume that ${\displaystyle O}$ has complete information on the system, and that ${\displaystyle O}$ can write down the wavefunction ${\displaystyle |\psi \rangle }$ describing it. At the same time, there is another observer ${\displaystyle O'}$, who is interested in the state of the entire ${\displaystyle O}$-${\displaystyle S}$ system, and ${\displaystyle O'}$ likewise has complete information.

To analyse this system formally, we consider a system ${\displaystyle S}$ which may take one of two states, which we shall designate ${\displaystyle |\uparrow \rangle }$ and ${\displaystyle |\downarrow \rangle }$, ket vectors in the Hilbert space ${\displaystyle H_{S}}$. Now, the observer ${\displaystyle O}$ wishes to make a measurement on the system. At time ${\displaystyle t_{1}}$, this observer may characterize the system as follows:

${\displaystyle |\psi \rangle =\alpha |\uparrow \rangle +\beta |\downarrow \rangle ,}$

where ${\displaystyle |\alpha |^{2}}$ and ${\displaystyle |\beta |^{2}}$ are probabilities of finding the system in the respective states, and obviously add up to 1. For our purposes here, we can assume that in a single experiment, the outcome is the eigenstate ${\displaystyle |\uparrow \rangle }$ (but this can be substituted throughout, mutatis mutandis, by ${\displaystyle |\downarrow \rangle }$). So, we may represent the sequence of events in this experiment, with observer ${\displaystyle O}$ doing the observing, as follows:

${\displaystyle {\begin{matrix}t_{1}&\rightarrow &t_{2}\\\alpha |\uparrow \rangle +\beta |\downarrow \rangle &\rightarrow &|\uparrow \rangle .\end{matrix}}}$

This is observer ${\displaystyle O}$'s description of the measurement event. Now, any measurement is also a physical interaction between two or more systems. Accordingly, we can consider the tensor product Hilbert space ${\displaystyle H_{S}\otimes H_{O}}$, where ${\displaystyle H_{O}}$ is the Hilbert space inhabited by state vectors describing ${\displaystyle O}$. If the initial state of ${\displaystyle O}$ is ${\displaystyle |{\text{init}}\rangle }$, some degrees of freedom in ${\displaystyle O}$ become correlated with the state of ${\displaystyle S}$ after the measurement, and this correlation can take one of two values: ${\displaystyle |O_{\uparrow }\rangle }$ or ${\displaystyle |O_{\downarrow }\rangle }$ where the direction of the arrows in the subscripts corresponds to the outcome of the measurement that ${\displaystyle O}$ has made on ${\displaystyle S}$. If we now consider the description of the measurement event by the other observer, ${\displaystyle O'}$, who describes the combined ${\displaystyle S+O}$ system, but does not interact with it, the following gives the description of the measurement event according to ${\displaystyle O'}$, from the linearity inherent in the quantum formalism:

${\displaystyle {\begin{matrix}t_{1}&\rightarrow &t_{2}\\\left(\alpha |\uparrow \rangle +\beta |\downarrow \rangle \right)\otimes |init\rangle &\rightarrow &\alpha |\uparrow \rangle \otimes |O_{\uparrow }\rangle +\beta |\downarrow \rangle \otimes |O_{\downarrow }\rangle .\end{matrix}}}$

Thus, on the assumption (see hypothesis 2 below) that quantum mechanics is complete, the two observers ${\displaystyle O}$ and ${\displaystyle O'}$ give different but equally correct accounts of the events ${\displaystyle t_{1}\rightarrow t_{2}}$.

## Central principles

### Observer-dependence of state

According to ${\displaystyle O}$, at ${\displaystyle t_{2}}$, the system ${\displaystyle S}$ is in a determinate state, namely spin up. And, if quantum mechanics is complete, then so is his description. But, for ${\displaystyle O'}$, ${\displaystyle S}$ is not uniquely determinate, but is rather entangled with the state of ${\displaystyle O}$ — note that his description of the situation at ${\displaystyle t_{2}}$ is not factorisable no matter what basis chosen. But, if quantum mechanics is complete, then the description that ${\displaystyle O'}$ gives is also complete.

Thus the standard mathematical formulation of quantum mechanics allows different observers to give different accounts of the same sequence of events. There are many ways to overcome this perceived difficulty. It could be described as an epistemic limitation — observers with a full knowledge of the system, we might say, could give a complete and equivalent description of the state of affairs, but that obtaining this knowledge is impossible in practice. But whom? What makes ${\displaystyle O}$'s description better than that of ${\displaystyle O'}$, or vice versa? Alternatively, we could claim that quantum mechanics is not a complete theory, and that by adding more structure we could arrive at a universal description (the troubled hidden variables approach). Yet another option is to give a preferred status to a particular observer or type of observer, and assign the epithet of correctness to their description alone. This has the disadvantage of being ad hoc, since there are no clearly defined or physically intuitive criteria by which this super-observer ("who can observe all possible sets of observations by all observers over the entire universe"[11]) ought to be chosen.

RQM, however, takes the point illustrated by this problem at face value. Instead of trying to modify quantum mechanics to make it fit with prior assumptions that we might have about the world, Rovelli says that we should modify our view of the world to conform to what amounts to our best physical theory of motion.[12] Just as forsaking the notion of absolute simultaneity helped clear up the problems associated with the interpretation of the Lorentz transformations, so many of the conundra associated with quantum mechanics dissolve, provided that the state of a system is assumed to be observer-dependent — like simultaneity in Special Relativity. This insight follows logically from the two main hypotheses which inform this interpretation:

• Hypothesis 1: the equivalence of systems. There is no a priori distinction that should be drawn between quantum and macroscopic systems. All systems are, fundamentally, quantum systems.
• Hypothesis 2: the completeness of quantum mechanics. There are no hidden variables or other factors which may be appropriately added to quantum mechanics, in light of current experimental evidence.

Thus, if a state is to be observer-dependent, then a description of a system would follow the form "system S is in state x with reference to observer O" or similar constructions, much like in relativity theory. In RQM it is meaningless to refer to the absolute, observer-independent state of any system.

### Information and correlation

It is generally well established that any quantum mechanical measurement can be reduced to a set of yes/no questions or bits that are either 1 or 0.[citation needed] RQM makes use of this fact to formulate the state of a quantum system (relative to a given observer!) in terms of the physical notion of information developed by Claude Shannon. Any yes/no question can be described as a single bit of information. This should not be confused with the idea of a qubit from quantum information theory, because a qubit can be in a superposition of values, whilst the "questions" of RQM are ordinary binary variables.

Any quantum measurement is fundamentally a physical interaction between the system being measured and some form of measuring apparatus. By extension, any physical interaction may be seen to be a form of quantum measurement, as all systems are seen as quantum systems in RQM. A physical interaction is seen as establishing a correlation between the system and the observer, and this correlation is what is described and predicted by the quantum formalism.

But, Rovelli points out, this form of correlation is precisely the same as the definition of information in Shannon's theory. Specifically, an observer O observing a system S will, after measurement, have some degrees of freedom correlated with those of S. The amount of this correlation is given by log2k bits, where k is the number of possible values which this correlation may take — the number of "options" there are.

### All systems are quantum systems

All physical interactions are, at bottom, quantum interactions, and must ultimately be governed by the same rules. Thus, an interaction between two particles does not, in RQM, differ fundamentally from an interaction between a particle and some "apparatus". There is no true wave collapse, in the sense in which it occurs in the Copenhagen interpretation.

Because "state" is expressed in RQM as the correlation between two systems, there can be no meaning to "self-measurement". If observer ${\displaystyle O}$ measures system ${\displaystyle S}$, ${\displaystyle S}$'s "state" is represented as a correlation between ${\displaystyle O}$ and ${\displaystyle S}$. ${\displaystyle O}$ itself cannot say anything with respect to its own "state", because its own "state" is defined only relative to another observer, ${\displaystyle O'}$. If the ${\displaystyle S+O}$ compound system does not interact with any other systems, then it will possess a clearly defined state relative to ${\displaystyle O'}$. However, because ${\displaystyle O}$'s measurement of ${\displaystyle S}$ breaks its unitary evolution with respect to ${\displaystyle O}$, ${\displaystyle O}$ will not be able to give a full description of the ${\displaystyle S+O}$ system (since it can only speak of the correlation between ${\displaystyle S}$ and itself, not its own behaviour). A complete description of the ${\displaystyle (S+O)+O'}$ system can only be given by a further, external observer, and so forth.

Taking the model system discussed above, if ${\displaystyle O'}$ has full information on the ${\displaystyle S+O}$ system, it will know the Hamiltonians of both ${\displaystyle S}$ and ${\displaystyle O}$, including the interaction Hamiltonian. Thus, the system will evolve entirely unitarily (without any form of collapse) relative to ${\displaystyle O'}$, if ${\displaystyle O}$ measures ${\displaystyle S}$. The only reason that ${\displaystyle O}$ will perceive a "collapse" is because ${\displaystyle O}$ has incomplete information on the system (specifically, ${\displaystyle O}$ does not know its own Hamiltonian, and the interaction Hamiltonian for the measurement).

## Consequences and implications

### Coherence

In our system above, ${\displaystyle O'}$ may be interested in ascertaining whether or not the state of ${\displaystyle O}$ accurately reflects the state of ${\displaystyle S}$. We can draw up for ${\displaystyle O'}$ an operator, ${\displaystyle M}$, which is specified as:

${\displaystyle M\left(|\uparrow \rangle \otimes |O_{\uparrow }\rangle \right)=|\uparrow \rangle \otimes |O_{\uparrow }\rangle }$
${\displaystyle M\left(|\uparrow \rangle \otimes |O_{\downarrow }\rangle \right)=0}$
${\displaystyle M\left(|\downarrow \rangle \otimes |O_{\uparrow }\rangle \right)=0}$
${\displaystyle M\left(|\downarrow \rangle \otimes |O_{\downarrow }\rangle \right)=|\downarrow \rangle \otimes |O_{\downarrow }\rangle }$

with an eigenvalue of 1 meaning that ${\displaystyle O}$ indeed accurately reflects the state of ${\displaystyle S}$. So there is a 0 probability of ${\displaystyle O}$ reflecting the state of ${\displaystyle S}$ as being ${\displaystyle |\uparrow \rangle }$ if it is in fact ${\displaystyle |\downarrow \rangle }$, and so forth. The implication of this is that at time ${\displaystyle t_{2}}$, ${\displaystyle O'}$ can predict with certainty that the ${\displaystyle S+O}$ system is in some eigenstate of ${\displaystyle M}$, but cannot say which eigenstate it is in, unless ${\displaystyle O'}$ itself interacts with the ${\displaystyle S+O}$ system.

An apparent paradox arises when one considers the comparison, between two observers, of the specific outcome of a measurement. In the problem of the observer observed section above, let us imagine that the two experiments want to compare results. It is obvious that if the observer ${\displaystyle O'}$ has the full Hamiltonians of both ${\displaystyle S}$ and ${\displaystyle O}$, he will be able to say with certainty that at time ${\displaystyle t_{2}}$, ${\displaystyle O}$ has a determinate result for ${\displaystyle S}$'s spin, but he will not be able to say what ${\displaystyle O}$'s result is without interaction, and hence breaking the unitary evolution of the compound system (because he doesn't know his own Hamiltonian). The distinction between knowing "that" and knowing "what" is a common one in everyday life: everyone knows that the weather will be like something tomorrow, but no-one knows exactly what the weather will be like.

But, let us imagine that ${\displaystyle O'}$ measures the spin of ${\displaystyle S}$, and finds it to have spin down (and note that nothing in the analysis above precludes this from happening). What happens if he talks to ${\displaystyle O}$, and they compare the results of their experiments? ${\displaystyle O}$, it will be remembered, measured a spin up on the particle. This would appear to be paradoxical: the two observers, surely, will realise that they have disparate results.

However, this apparent paradox only arises as a result of the question being framed incorrectly: as long as we presuppose an "absolute" or "true" state of the world, this would, indeed, present an insurmountable obstacle for the relational interpretation. However, in a fully relational context, there is no way in which the problem can even be coherently expressed. The consistency inherent in the quantum formalism, exemplified by the "M-operator" defined above, guarantees that there will be no contradictions between records. The interaction between ${\displaystyle O'}$ and whatever he chooses to measure, be it the ${\displaystyle S+O}$ compound system or ${\displaystyle O}$ and ${\displaystyle S}$ individually, will be a physical interaction, a quantum interaction, and so a complete description of it can only be given by a further observer ${\displaystyle O''}$, who will have a similar "M-operator" guaranteeing coherency, and so on out. In other words, a situation such as that described above cannot violate any physical observation, as long as the physical content of quantum mechanics is taken to refer only to relations.

### Relational networks

An interesting implication of RQM arises when we consider that interactions between material systems can only occur within the constraints prescribed by Special Relativity, namely within the intersections of the light cones of the systems: when they are spatiotemporally contiguous, in other words. Relativity tells us that objects have location only relative to other objects. By extension, a network of relations could be built up based on the properties of a set of systems, which determines which systems have properties relative to which others, and when (since properties are no longer well defined relative to a specific observer after unitary evolution breaks down for that observer). On the assumption that all interactions are local (which is backed up by the analysis of the EPR paradox presented below), one could say that the ideas of "state" and spatiotemporal contiguity are two sides of the same coin: spacetime location determines the possibility of interaction, but interactions determine spatiotemporal structure. The full extent of this relationship, however, has not yet fully been explored.

### RQM and quantum cosmology

The universe is the sum total of everything in existence with any possibility of direct or indirect interaction with a local observer. A (physical) observer outside of the universe would require physically breaking of gauge invariance,[13] and a concomitant alteration in the mathematical structure of gauge-invariance theory.

Similarly, RQM conceptually forbids the possibility of an external observer. Since the assignment of a quantum state requires at least two "objects" (system and observer), which must both be physical systems, there is no meaning in speaking of the "state" of the entire universe. This is because this state would have to be ascribed to a correlation between the universe and some other physical observer, but this observer in turn would have to form part of the universe. As was discussed above, it is not possible for an object to contain a complete specification of itself. Following the idea of relational networks above, an RQM-oriented cosmology would have to account for the universe as a set of partial systems providing descriptions of one another. The exact nature of such a construction remains an open question.

## Relationship with other interpretations

The only group of interpretations of quantum mechanics with which RQM is almost completely incompatible is that of hidden variables theories. RQM shares some deep similarities with other views, but differs from them all to the extent to which the other interpretations do not accord with the "relational world" put forward by RQM.

### Copenhagen interpretation

RQM is, in essence, quite similar to the Copenhagen interpretation, but with an important difference. In the Copenhagen interpretation, the macroscopic world is assumed to be intrinsically classical in nature, and wave function collapse occurs when a quantum system interacts with macroscopic apparatus. In RQM, any interaction, be it micro or macroscopic, causes the linearity of Schrödinger evolution to break down. RQM could recover a Copenhagen-like view of the world by assigning a privileged status (not dissimilar to a preferred frame in relativity) to the classical world. However, by doing this one would lose sight of the key features that RQM brings to our view of the quantum world.

### Hidden variables theories

Bohm's interpretation of QM does not sit well with RQM. One of the explicit hypotheses in the construction of RQM is that quantum mechanics is a complete theory, that is it provides a full account of the world. Moreover, the Bohmian view seems to imply an underlying, "absolute" set of states of all systems, which is also ruled out as a consequence of RQM.

We find a similar incompatibility between RQM and suggestions such as that of Penrose, which postulate that some process (in Penrose's case, gravitational effects) violate the linear evolution of the Schrödinger equation for the system.

### Relative-state formulation

The many-worlds family of interpretations (MWI) shares an important feature with RQM, that is, the relational nature of all value assignments (that is, properties). Everett, however, maintains that the universal wavefunction gives a complete description of the entire universe, while Rovelli argues that this is problematic, both because this description is not tied to a specific observer (and hence is "meaningless" in RQM), and because RQM maintains that there is no single, absolute description of the universe as a whole, but rather a net of inter-related partial descriptions.

### Consistent histories approach

In the consistent histories approach to QM, instead of assigning probabilities to single values for a given system, the emphasis is given to sequences of values, in such a way as to exclude (as physically impossible) all value assignments which result in inconsistent probabilities being attributed to observed states of the system. This is done by means of ascribing values to "frameworks", and all values are hence framework-dependent.

RQM accords perfectly well with this view. However, the consistent histories approach does not give a full description of the physical meaning of framework-dependent value (that is it does not account for how there can be "facts" if the value of any property depends on the framework chosen). By incorporating the relational view into this approach, the problem is solved: RQM provides the means by which the observer-independent, framework-dependent probabilities of various histories are reconciled with observer-dependent descriptions of the world.

## EPR and quantum non-locality

The EPR thought experiment, performed with electrons. A radioactive source (center) sends electrons in a singlet state toward two spacelike separated observers, Alice (left) and Bob (right), who can perform spin measurements. If Alice measures spin up on her electron, Bob will measure spin down on his, and vice versa.

RQM provides an unusual solution to the EPR paradox. Indeed, it manages to dissolve the problem altogether, inasmuch as there is no superluminal transportation of information involved in a Bell test experiment: the principle of locality is preserved inviolate for all observers.

### The problem

In the EPR thought experiment, a radioactive source produces two electrons in a singlet state, meaning that the sum of the spin on the two electrons is zero. These electrons are fired off at time ${\displaystyle t_{1}}$ towards two spacelike separated observers, Alice and Bob, who can perform spin measurements, which they do at time ${\displaystyle t_{2}}$. The fact that the two electrons are a singlet means that if Alice measures z-spin up on her electron, Bob will measure z-spin down on his, and vice versa: the correlation is perfect. If Alice measures z-axis spin, and Bob measures the orthogonal y-axis spin, however, the correlation will be zero. Intermediate angles give intermediate correlations in a way that, on careful analysis, proves inconsistent with the idea that each particle has a definite, independent probability of producing the observed measurements (the correlations violate Bell's inequality).

This subtle dependence of one measurement on the other holds even when measurements are made simultaneously and a great distance apart, which gives the appearance of a superluminal communication taking place between the two electrons. Put simply, how can Bob's electron "know" what Alice measured on hers, so that it can adjust its own behavior accordingly?

### Relational solution

In RQM, an interaction between a system and an observer is necessary for the system to have clearly defined properties relative to that observer. Since the two measurement events take place at spacelike separation, they do not lie in the intersection of Alice's and Bob's light cones. Indeed, there is no observer who can instantaneously measure both electrons' spin.

The key to the RQM analysis is to remember that the results obtained on each "wing" of the experiment only become determinate for a given observer once that observer has interacted with the other observer involved. As far as Alice is concerned, the specific results obtained on Bob's wing of the experiment are indeterminate for her, although she will know that Bob has a definite result. In order to find out what result Bob has, she has to interact with him at some time ${\displaystyle t_{3}}$ in their future light cones, through ordinary classical information channels.[14]

The question then becomes one of whether the expected correlations in results will appear: will the two particles behave in accordance with the laws of quantum mechanics? Let us denote by ${\displaystyle M_{A}(\alpha )}$ the idea that the observer ${\displaystyle A}$ (Alice) measures the state of the system ${\displaystyle \alpha }$ (Alice's particle).

So, at time ${\displaystyle t_{2}}$, Alice knows the value of ${\displaystyle M_{A}(\alpha )}$: the spin of her particle, relative to herself. But, since the particles are in a singlet state, she knows that

${\displaystyle M_{A}(\alpha )+M_{A}(\beta )=0,}$

and so if she measures her particle's spin to be ${\displaystyle \sigma }$, she can predict that Bob's particle (${\displaystyle \beta }$) will have spin ${\displaystyle -\sigma }$. All this follows from standard quantum mechanics, and there is no "spooky action at a distance" yet. From the "coherence-operator" discussed above, Alice also knows that if at ${\displaystyle t_{3}}$ she measures Bob's particle and then measures Bob (that is asks him what result he got) — or vice versa — the results will be consistent:

${\displaystyle M_{A}(B)=M_{A}(\beta )}$

Finally, if a third observer (Charles, say) comes along and measures Alice, Bob, and their respective particles, he will find that everyone still agrees, because his own "coherence-operator" demands that

${\displaystyle M_{C}(A)=M_{C}(\alpha )}$ and ${\displaystyle M_{C}(B)=M_{C}(\beta )}$

while knowledge that the particles were in a singlet state tells him that

${\displaystyle M_{C}(\alpha )+M_{C}(\beta )=0.}$

Thus the relational interpretation, by shedding the notion of an "absolute state" of the system, allows for an analysis of the EPR paradox which neither violates traditional locality constraints, nor implies superluminal information transfer, since we can assume that all observers are moving at comfortable sub-light velocities. And, most importantly, the results of every observer are in full accordance with those expected by conventional quantum mechanics.

## Derivation

A promising feature of this interpretation is that RQM offers the possibility of being derived from a small number of axioms, or postulates based on experimental observations. Rovelli's derivation of RQM uses three fundamental postulates. However, it has been suggested that it may be possible to reformulate the third postulate into a weaker statement, or possibly even do away with it altogether.[15] The derivation of RQM parallels, to a large extent, quantum logic. The first two postulates are motivated entirely by experimental results, while the third postulate, although it accords perfectly with what we have discovered experimentally, is introduced as a means of recovering the full Hilbert space formalism of quantum mechanics from the other two postulates. The two empirical postulates are:

• Postulate 1: there is a maximum amount of relevant information that may be obtained from a quantum system.
• Postulate 2: it is always possible to obtain new information from a system.

We let ${\displaystyle W\left(S\right)}$ denote the set of all possible questions that may be "asked" of a quantum system, which we shall denote by ${\displaystyle Q_{i}}$, ${\displaystyle i\in W}$. We may experimentally find certain relations between these questions: ${\displaystyle \left\{\land ,\lor ,\neg ,\supset ,\bot \right\}}$, corresponding to {intersection, orthogonal sum, orthogonal complement, inclusion, and orthogonality} respectively, where ${\displaystyle Q_{1}\bot Q_{2}\equiv Q_{1}\supset \neg Q_{2}}$.

### Structure

From the first postulate, it follows that we may choose a subset ${\displaystyle Q_{c}^{(i)}}$ of ${\displaystyle N}$ mutually independent questions, where ${\displaystyle N}$ is the number of bits contained in the maximum amount of information. We call such a question ${\displaystyle Q_{c}^{(i)}}$ a complete question. The value of ${\displaystyle Q_{c}^{(i)}}$ can be expressed as an N-tuple sequence of binary valued numerals, which has ${\displaystyle 2^{N}=k}$ possible permutations of "0" and "1" values. There will also be more than one possible complete question. If we further assume that the relations ${\displaystyle \left\{\land ,\lor \right\}}$ are defined for all ${\displaystyle Q_{i}}$, then ${\displaystyle W\left(S\right)}$ is an orthomodular lattice, while all the possible unions of sets of complete questions form a Boolean algebra with the ${\displaystyle Q_{c}^{(i)}}$ as atoms.[16]

The second postulate governs the event of further questions being asked by an observer ${\displaystyle O_{1}}$ of a system ${\displaystyle S}$, when ${\displaystyle O_{1}}$ already has a full complement of information on the system (an answer to a complete question). We denote by ${\displaystyle p\left(Q|Q_{c}^{(j)}\right)}$ the probability that a "yes" answer to a question ${\displaystyle Q}$ will follow the complete question ${\displaystyle Q_{c}^{(j)}}$. If ${\displaystyle Q}$ is independent of ${\displaystyle Q_{c}^{(j)}}$, then ${\displaystyle p=0.5}$, or it might be fully determined by ${\displaystyle Q_{c}^{(j)}}$, in which case ${\displaystyle p=1}$. There is also a range of intermediate possibilities, and this case is examined below.

If the question that ${\displaystyle O_{1}}$ wants to ask the system is another complete question, ${\displaystyle Q_{b}^{(i)}}$, the probability ${\displaystyle p^{ij}=p\left(Q_{b}^{(i)}|Q_{c}^{(j)}\right)}$ of a "yes" answer has certain constraints upon it:

1. ${\displaystyle 0\leq p^{ij}\leq 1,\ }$
2. ${\displaystyle \sum _{i}p^{ij}=1,\ }$
3. ${\displaystyle \sum _{j}p^{ij}=1.\ }$

The three constraints above are inspired by the most basic of properties of probabilities, and are satisfied if

${\displaystyle p^{ij}=\left|U^{ij}\right|^{2}}$,

where ${\displaystyle U^{ij}}$ is a unitary matrix.

• Postulate 3 If ${\displaystyle b}$ and ${\displaystyle c}$ are two complete questions, then the unitary matrix ${\displaystyle U_{bc}}$ associated with their probability described above satisfies the equality ${\displaystyle U_{cd}=U_{cb}U_{bd}}$, for all ${\displaystyle b,c}$ and ${\displaystyle d}$.

This third postulate implies that if we set a complete question ${\displaystyle |Q_{c}^{(i)}\rangle }$ as a basis vector in a complex Hilbert space, we may then represent any other question ${\displaystyle |Q_{b}^{(j)}\rangle }$ as a linear combination:

${\displaystyle |Q_{b}^{(j)}\rangle =\sum _{i}U_{bc}^{ij}|Q_{c}^{(i)}\rangle .}$

And the conventional probability rule of quantum mechanics states that if two sets of basis vectors are in the relation above, then the probability ${\displaystyle p^{ij}}$ is

${\displaystyle p^{ij}=|\langle Q_{c}^{(i)}|Q_{b}^{(j)}\rangle |^{2}=|U_{bc}^{ij}|^{2}.}$

### Dynamics

The Heisenberg picture of time evolution accords most easily with RQM. Questions may be labelled by a time parameter ${\displaystyle t\rightarrow Q(t)}$, and are regarded as distinct if they are specified by the same operator but are performed at different times. Because time evolution is a symmetry in the theory (it forms a necessary part of the full formal derivation of the theory from the postulates), the set of all possible questions at time ${\displaystyle t_{2}}$ is isomorphic to the set of all possible questions at time ${\displaystyle t_{1}}$. It follows, by standard arguments in quantum logic, from the derivation above that the orthomodular lattice ${\displaystyle W(S)}$ has the structure of the set of linear subspaces of a Hilbert space, with the relations between the questions corresponding to the relations between linear subspaces.

It follows that there must be a unitary transformation ${\displaystyle U\left(t_{2}-t_{1}\right)}$ that satisfies:

${\displaystyle Q(t_{2})=U\left(t_{2}-t_{1}\right)Q(t_{1})U^{-1}\left(t_{2}-t_{1}\right)}$

and

${\displaystyle U\left(t_{2}-t_{1}\right)=\exp({-i\left(t_{2}-t_{1}\right)H})}$

where ${\displaystyle H}$ is the Hamiltonian, a self-adjoint operator on the Hilbert space and the unitary matrices are an abelian group.

## Notes

1. ^ Wheeler (1990): pg. 3
2. ^ Rovelli, C. (1996), "Relational quantum mechanics", International Journal of Theoretical Physics, 35: 1637–1678.
3. ^ Rovelli (1996): pg. 2
4. ^ Smolin (1995)
5. ^ Crane (1993)
6. ^ Laudisa (2001)
7. ^ Rovelli & Smerlak (2006)
8. ^ Mermin, N.D. (1996, 1998).
9. ^ mikhailfranco (Nov 2008) web comment.
10. ^ Garret, R. (Jan 2011) "The Quantum Conspiracy: What Popularizers Of QM Don't Want You To Know" (slides, video),
11. ^ Page, Don N., "Insufficiency of the quantum state for deducing observational probabilities", Physics Letters B, Volume 678, Issue 1, 6 July 2009, 41-44.
12. ^ Rovelli (1996): pg. 16
13. ^ Smolin (1995), pg. 13
14. ^ Bitbol (1983)
15. ^ Rovelli (1996): pg. 14
16. ^ Rovelli (1996): pg. 13

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