Relational quantum mechanics (RQM) is an interpretation of quantum mechanics which treats the state of a quantum system as being relational, 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,[1] 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.[2]
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".[3]
The essential idea behind RQM is that different observers may give different accurate accounts of the same system. For example, to one observer, a system is in a single, "collapsed" eigenstate. To a second observer, the same system is in a superposition of two or more states and the first observer is in a correlated superposition of two or more states. RQM argues that this is a complete picture of the world because the notion of "state" is always relative to some observer. There is no privileged, "real" account. 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. The terms "observer" and "observed" apply to any arbitrary system, microscopic or macroscopic. The classical limit is a consequence of aggregate systems of very highly correlated subsystems.A "measurement event" is thus described as an ordinary physical interaction where two systems become correlated to some degree with respect to each other.
Rovelli criticizes describing this as a form of "observer-dependence" which suggests reality depends upon the presence of a conscious observer, when his point is instead that reality is relational and thus the state of a system can be described even in relation to any physical object and not necessarily a human observer.[4]
The proponents of the relational interpretation argue that this approach resolves some of the traditional interpretational difficulties with quantum mechanics. By giving up our preconception of a global privileged state, issues around the measurement problem and local realism are resolved.
In 2020, Carlo Rovelli published an account of the main ideas of the relational interpretation in his popular book Helgoland, which was published in an English translation in 2021 as Helgoland: Making Sense of the Quantum Revolution.[5]
Relational quantum mechanics arose from a comparison of the quandaries posed by the interpretations of quantum mechanics with those resulting from Lorentz transformations prior to the development of special relativity. Rovelli suggested that just as pre-relativistic interpretations of Lorentz's equations were complicated by incorrectly assuming an observer-independent time exists, a similarly incorrect assumption frustrates attempts to make sense of the quantum formalism. The assumption rejected by relational quantum mechanics is the existence of an observer-independent state of a system.[6]
The idea has been expanded upon by Lee Smolin[7] and Louis Crane,[8] 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.[9] [10]
This problem was initially discussed in detail in Everett's thesis, The Theory of the Universal Wavefunction. Consider observer
O
S
O
O
|\psi\rangle
O'
O
S
O'
To analyse this system formally, we consider a system
S
|{\uparrow}\rangle
|\downarrow\rangle
HS
O
t1
|\psi\rangle=\alpha|{\uparrow}\rangle+\beta|{\downarrow}\rangle,
where
|\alpha|2
|\beta|2
|{\uparrow}\rangle
|{\downarrow}\rangle
O
\begin{matrix}t1& → &t2\\ \alpha|{\uparrow}\rangle+\beta|{\downarrow}\rangle& → &|{\uparrow}\rangle. \end{matrix}
This is the description of the measurement event given by observer
O
HS ⊗ HO
HO
O
O
|init\rangle
O
S
|O\uparrow\rangle
|O\downarrow\rangle
O
S
O'
S+O
O'
\begin{matrix} t1& → &t2\\ \left(\alpha|{\uparrow}\rangle+\beta|{\downarrow}\rangle\right) ⊗ |init\rangle& → &\alpha|{\uparrow}\rangle ⊗ |O\uparrow\rangle+\beta|{\downarrow}\rangle ⊗ |O\downarrow\rangle. \end{matrix}
Thus, on the assumption (see hypothesis 2 below) that quantum mechanics is complete, the two observers
O
O'
t1 → t2
Note that the above scenario is directly linked to Wigner's Friend thought experiment, which serves as a prime example when understanding different interpretations of quantum theory.
According to
O
t2
S
O'
S
O
t2
O'
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
O
O'
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 conundrums 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:
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.
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. 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 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 some interpretations.
Because "state" is expressed in RQM as the correlation between two systems, there can be no meaning to "self-measurement". If observer
O
S
S
O
S
O
O'
S+O
O'
O
S
O
O
S+O
S
(S+O)+O'
Taking the model system discussed above, if
O'
S+O
S
O
O'
O
S
O
O
O
In our system above,
O'
O
S
O'
M
M\left(|{\uparrow}\rangle ⊗ |O\uparrow\rangle\right)=|{\uparrow}\rangle ⊗ |O\uparrow\rangle
M\left(|{\uparrow}\rangle ⊗ |O\downarrow\rangle\right)=0
M\left(|{\downarrow}\rangle ⊗ |O\uparrow\rangle\right)=0
M\left(|{\downarrow}\rangle ⊗ |O\downarrow\rangle\right)=|{\downarrow}\rangle ⊗ |O\downarrow\rangle
O
S
O
S
|{\uparrow}\rangle
|{\downarrow}\rangle
t2
O'
S+O
M
O'
S+O
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
O'
S
O
t2
O
S
O
But, let us imagine that
O'
S
O
O
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
O'
S+O
O
S
O''
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.
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. Such a construction was developed in particular by Francesca Vidotto .[14]
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.
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.
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.
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 interrelated partial descriptions.
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.
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.
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
t1
t2
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?
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
t3
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
MA(\alpha)
A
\alpha
So, at time
t2
MA(\alpha)
MA(\alpha)+MA(\beta)=0,
and so if she measures her particle's spin to be
\sigma
\beta
-\sigma
t3
MA(B)=MA(\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
MC(A)=MC(\alpha)
MC(B)=MC(\beta)
while knowledge that the particles were in a singlet state tells him that
MC(\alpha)+MC(\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.
Whether or not this account of locality is successful has been a matter of debate.[16]
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.[17] 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:
We let
W\left(S\right)
Qi
i\inW
\left\{\land,\lor,\neg,\supset,\bot\right\}
Q1\botQ2\equivQ1\supset\negQ2
From the first postulate, it follows that we may choose a subset
| (i) | |
| Q | |
| c |
N
N
| (i) | |
| Q | |
| c |
| (i) | |
| Q | |
| c |
2N=k
\left\{\land,\lor\right\}
Qi
W\left(S\right)
| (i) | |
| Q | |
| c |
The second postulate governs the event of further questions being asked by an observer
O1
S
O1
| (j) | |
| p\left(Q|Q | |
| c |
\right)
Q
| (j) | |
| Q | |
| c |
Q
| (j) | |
| Q | |
| c |
p=0.5
| (j) | |
| Q | |
| c |
p=1
If the question that
O1
| (i) | |
| Q | |
| b |
pij
| (i) | |
| =p\left(Q | |
| b |
| (j) | |
| |Q | |
| c |
\right)
1.
0\leqpij\leq1,
2.
\sumipij=1,
3.
\sumjpij=1.
The three constraints above are inspired by the most basic of properties of probabilities, and are satisfied if
pij=\left|Uij\right|2
where
Uij
b
c
Ubc
Ucd=UcbUbd
b,c
d
This third postulate implies that if we set a complete question
| (i) | |
| |Q | |
| c |
\rangle
| (j) | |
| |Q | |
| b |
\rangle
| (j) | |
| |Q | |
| b |
\rangle=\sumi
| ij | |
| U | |
| bc |
| (i) | |
| |Q | |
| c |
\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
pij
pij=|\langle
| (i) | |
| Q | |
| c |
|
| (j) | |
| Q | |
| b |
\rangle|2=
| ij | |
| |U | |
| bc |
|2.
The Heisenberg picture of time evolution accords most easily with RQM. Questions may be labelled by a time parameter
t → Q(t)
t2
t1
W(S)
U\left(t2-t1\right)
Q(t2)=U\left(t2-t1\right)Q(t1)U-1\left(t2-t1\right)
and
U\left(t2-t1\right)=\exp({-i\left(t2-t1\right)H})
where
H
The question is whether RQM denies any objective reality, or otherwise stated: there is only a subjectively knowable reality. Rovelli limits the scope of this claim by stating that RQM relates to the variables of a physical system and not to constant, intrinsic properties, such as the mass and charge of an electron.[19] Indeed, mechanics in general only predicts the behavior of a physical system under various conditions. In classical mechanics this behavior is mathematically represented in a phase space with certain degrees of freedom; in quantum mechanics this is a state space, mathematically represented as a multidimensional complex Hilbert space, in which the dimensions correspond to the above variables.Dorato,[20] however, argues that all intrinsic properties of a physical system, including mass and charge, are only knowable in a subjective interaction between the observer and the physical system. The unspoken thought behind this is that intrinsic properties are essentially quantum mechanical properties as well.