The mission of the Internet Science Education Project is to popularize important advanced ideas in theoretical physics on the World Wide Web using multimedia when the bandwidth using cable modems becomes common place. For now we are restricted mostly to text. Lee Smolin's work is the first in the new series "New Physics Reviews". Eventually the technical terms here, like "diffeomorphism", will be hyperlinked to background information. Many students using the web do not have the required software to read the electronic post-script files that the original preprints come in. The level here will hopefully be simpler and more intuitively physical than the similar posts by John Baez to which we will link when appropriate.
In "Time and measurement in quantum cosmology" (9301016), Lee Smolin uses Bohr's Copenhagen interpretation in an attempt to construct nonperturbative diffeomorphism-invariant physical operators, states, and inner products that describe clocks and measuring rods relative to a physical frame of reference. Everything is inside the universe. Smolin says that Gell-Mann's et-al claim that one must give up Bohr's interpretation to do quantum cosmology is wrong. Smolin's approach is operational and does not require decoherent consistent histories, Everett relative states or Bohm super quantum potentials, though these may add additional insights later on. Smolin says there is a meaning to singularities and nonunitary loss of information at the nonperturbative level where the gravitational degrees of freedom are quantized with no classical background metric. I wonder if the nonperturbative diffeomorphic invariant approach suffices to give the self-organizing post-quantum backactivity that I put in adhoc in the low energy flat spacetime limit of living carbon-based organisms with a "mental" collective mode isolated from random environmental decoherence? This would fit Penrose's general idea of linking moments of consciousness to self-collapses from the quantum gravity level - even though that is a strongly counter-intuitive idea because of the enormous difference in scales.
A quantum reference system, for canonical quantum gravity in the loop representation, has relative spatial separations fixed by the configurations of matter fields. The regularized observables are finite and spatially-diffeomorphic invariant. The spatial 3- geometry is compact with fixed topology and differential structure. The spatial 3-geometry is the basic "beable" of quantum gravity the way the "particle" is in Bohm's pilot-wave/hidden-variable picture of ordinary quantum mechanics in flat space time in the nonrelativistic low velocity and Newtonian weak gravity limit. You cannot measure things locally at a point. You can measure nonlocal things like areas, volumes and parallel transports. You need matter fields to label sets of open surfaces with boundaries in the 3-geometry. The matter fields can be scalar, antisymmetric tensor or abelian gauge. The matter fields are quantized in a "surface representation" where the states are functionals on the sets of open bounded surfaces. Each surface gives a Dirac bra for the matter field. There is an Abelian loop identity. N surfaces can be lumped together to form a many-surface bra. This matter system is coupled to gravity using the "loop representation". The gravity degrees of freedom are labeled by "loops". Impose and solve constraints for spatial diffeomorphism invariance. The quantum states of matter and curved space are functions of a countable number of "gauge" diffeomorphism equivalence classes of loops and N open surfaces with boundaries. Inner products in Hilbert space are not yet defined. That is, so far we only have Dirac bras with no kets. More precisly in Smolin's words:
"The space of is taken to be the space of functions ... if diffeomorphism invariant classes of loops and surfaces. The space of bras are defined to be linear maps from this space to the complex numbers."
There is still no inner product. That is, there is no isomorphism between the respective spaces of the bras and the kets. There are finite spatially diffeomorphism-invariant "area" observables in the loop representation that measure the gravitational field. The Dirac bras are eigenstates of these area operators. The eigenvalues are proportional to a positive unoriented intersection number of the loops with the surfaces. The second quantum gravity observable is the "Wilson loop" around the boundary of each open surface. Commutators for these two types of quantum-gravity observables are constructed.
Smolin claims a diffeomorphism-invariant description of the frame of reference and the measuring instruments. Each reference frame is identified with a topological configuration of the matter fields. Use a simplicial decomposition of the 3-geometry with an equal number N of edges and faces. For each such N-decomposition there is a subspace of the Hilbert space spanned by a basis in which the open bounded surfaces are 1-1 with the N faces of the simplicial decomposition. Therefore, they have the same topology. This is beautiful. The Bohr preparation is then specified by areas and Wilson parallel transports within this particular topology and Hilbert subspace. There are two maximal subsets of size N of mutually commuting observables. Each gives N numbers which is partial information about the spacetime geometry as a result of a measurement within the given quantum frame of reference. There is a classical limit for the quantum gravity d ynamical degrees of freedom. The output of measuring N area operators are N rational numbers multiplied by the square of the Planck length (10^-66 cm^2). These form a piecewise flat geometry called a "Regge manifold" of tetrahedra. The tetrahedral identities for individual tetrahedra in the Regge manifold need not be satisfied. One can have a loop intersecting only one face of a giving that face a finite area with other faces of zero area. Thus, there can be Regge manifolds that have tetrahedra with indefinite signatures. There can be more than one set of edges consistent with the areas. This gives a finite set of classical 3-geometries consistent with the quantum area measurement. Suppose we instead measure the generally incompatible Wilson loops for "self-dual" parallel transports. The spatial metric is therefore maximally uncertain consistent with the Heisenberg uncertainty principle. The parallel transports, giving information on the self-dual part of the spacetime curvature, are analogous to Fourier transforming from position X to momentum P space in ordinary quantum mechanics. A similar relation exists in going from object to image in classical far-field Fraunhofer diffraction useful in optical image processing. There is a dual graph to the spatial Regge manifold simplicial decomposition from the area observables. Each tetrahedron in the latter gives a vertex in the former. Each face in the latter gives an edge in the former. The four edges in the "momentum" dual graph from the four faces of an individual tetrahedron in the "position" Regge manifold, all meet at the same vertex in the dual graph. Each dual graph has a "distributional self-dual curvature" determined by N elements from the SL(2,C) algebra. A non-Abelian extension of Stoke's theorem shows that the moduli of these complex elements of SL(2,C) are determined from N ordinary complex numbers. This is analogous to a field of local phases in the ordinary phases of a flat spacetime quantum field theory with internal gauge symmetries. These local phases are points on a fiber. Note the SL(2,C) moduli appear to be ordinary complex numbers not real numbers because SL(1,C) uses the ordinary complex plane and SL(2,C) uses two such complex planes. A Chern-Simon connection field with the self-dual curvature as its source exists. Additional structure is required. It cannot depend on any preferred definite classical background spatial metric. Give an arbitrary specification of the faces of the self-dual graph. There is one face of this self-dual graph for each edge of a tetrahedron in the Regge manifold. This gives the connection field which resembles the Yang-Mills "potential" (i.e., eq 9, p. 13). So each measurement of the Wilson loops gives a classical geometry determined by N complex numbers which is a partial specification of a spatial Regge manifold. Smolin then seems to adds a discrete time ad hoc, so a self-dual edge becomes a self-dual face in a 4D extension of the original construction. The self-dual curvature is a distribution with support only on the faces. Smolin then builds the spatially diffeomorphism invariant inner product "bra-kets" for the Hilbert space of quantum gravity. The states that form the inner product must obey the Hamiltonian constraint which Smolin says is equivalent to the problem of how time emerges from the quantum gravity level. This requires adding additional operators to the algebra of area and Wilson loop observables to get a C* algebra. A diffeomorphic Kronecker delta is defined between different sets of open surfaces and loops. An adjoint map from kets to bras is constructed. An impotant case of this map is when the loops do not intersect each other.
Smolin uses the operational definition that time is what material clocks measure. The matter fields must allow the construction of clocks to transform the Hamiltonian constraint into an evolution equation for spatially diffeomorphic invariant observables relative to a material clock. The classical matter fields that describe the classical clock cannot be the same as those that describe the quantum reference frame (p.15) Smolin introduces a new classical massless scalar matter "clock field" T that defines the physical time. Its conjugate is an energy density. One need, therefore, to add new terms to the Hamiltonian and diffeomorphism constraints. There is a time displacement symmetry T -> T + constant. Choose a gauge fixing in which the spacelike slicing of spacetime corresponds to constant T. This determines the "lapse" in the cannonical ADM at the classical level. The gauge condition requires that all but one of an infinite number of Hamiltonian constraints are broken. The broken constraints must be solved to eliminate their conjugate fields. Thus, but one of the degrees of freedom in the energy density of the clock field are eliminated. The one that is kept independent is a global constant of the motion.
"... the gauge fixed theory is based on a phase space which consists of the original gravitational and matter phase space, to with the two conjugate degrees of freedom ... [global time and global energy density]... have been added. The diffeomorphism constrain remains the original one while there remains one Hamiltonian constraint ..." p. 18 Smolin then uses a method of Rovelli's to construct a one parameter gauge invariant family of spatially diffeomorphic invariant physical observables on the phase space called "evolving constants of motion" for each instant of global time. So far, this is all at the classical level. Smolin then extends this classical model to quantum theory. There are two methods one with gauge fixing and one without. Smolin gets a global time dependent Schrodinger equation with an operator square root which also requires a regularization procedure. These involve two "miracles" to really solve! (p. 20) But the two miracles may only be one because Smolin says that so far the only finite diffeomorphic invariant observables known all have operator square roots. Smolin needs to make a few optimistic assumptions to make progress such as W the space of solutions to the diffeomorphism constraints. Most important is that the global time evolution operator W for the non-clock fields is NOT Hermitian. This is consistent with my intrinsically non-unitary self-organizing novelty-generating "backactivity" idea. It may be a natural consequence of quantum gravity if Smolin is on the right track. It is backactivity which transforms the "rigid" orthodox quantum wave function into a "dynamic actor" depending directly on the path of its attached material beable in configuration space in Bohm's pilot-wave interpretation of physical reality. Unitarity means, says Smolin (p. 23) that all physical quantum gravity states exist for all values of the global time. This contradicts the Penrose-Hawking classical final singularity theorems for positive energy density in a compact space cosmology. Even apart from that there is at least one other reason why W cannot be Hermitian. In any case Smolin says that this is a dynamical problem for different gauge-fixed models of quantum gravity and that any such model that has gravitational collapse to a singularity cannot be unitary.
Smolin then considers a quantum gravity theory without gauge fixing. There is now the problem of the negative energy states which does not arise in the gauge-fixed theory which has only a first order derivative in the global time for the Hamiltonian constraint rather than a second order one. However Smolin attempts to construct a consistent positive energy constraint splitting the Hilbert space.
Smolin then gets into a discussion of preparation and measurement in quantum gravity.
A) The measurement theory must be fully 4D spacetime diffeomorphism invariant.
B) The frame of reference to determine "where" and "when" must be a dynamical component of matter and curvature dynamical degrees of freedom.
C) All possible measurements are partial.
D) The inner product obeys a reality condition at the initial time corresponding to the preparation.
So, Smolin's theory has two classes of diffeomorphic-invariant positive energy observables of non-clock fields, either in gauge-fixed or Dirac forms, at fixed global times. They are 1) measures of the areas of the simplices that define a given quantum frame of reference and 2) parallel transport around these simplices at a fixed global time. Preparation requires arrangement of the spatial reference system and synchronization of the material clocks. Unlike ordinary quantum mechanics "there is no basis of the diffeomorphism invariant space ... whose elements can be written as direct products of matter states and gravity states. Thus, the requirement of diffeomorphism invariance has entangled the various components of the whole system before any interactions have occurred." (p. 27) This, however, does not prevent preparation of the reference frame and clock matter fields. Project the physical states into appropriate subspaces. Smolin assumes that one can make a global synchronization over the whole spatial universe. Will the classical Hubble flow qualify? Does it obey diffeomorphic invariance. How does it relate to the topological simplices of Smolin's quantum frame? How do we move from the quantum frame to the classical frame? It had better qualify since it is well defined operationally. That is, moments of the same global time and at global rest correspond to the same cosmic blackbody photon absolute temperature with isotropy to one part in 10^5. This preferred cosmological global frame appears to be analogous to spontaneous broken symmetry in classical flat spacetime quantum theories where the lowest energy solution to a set of field equations does not have the full symmetry of those equations.
Smolin says that synchronizing the clocks requires that the inner product permit normalizable physical states. (p. 28) Preparing the spatial frame must clearly come after the fixed global time spacelike surfaces are specified. This process is apparently independent of specification of any quantum gravity physical state.
Smolin also must inquire whether the wave function epistemological (Bohr) or ontological (Everett, Bohm)? If the former, it is only subjective information in our minds and we use the pragmatic von Neumann projection postulate where "collapse" is subjective i.e., only a change of information in the mind, if the latter, it is objective i.e., "out there" independent of our mind, and we use either Everett's (many worlds) "relative state" or Bohm's objective "pilot-wave" acting on an attached material beable. Observables are made from beables. Penrose's and Stapp's post-quantum theories have a physical collapse absent in the Bohr theory. (p. 29) The problem in special relativity then is, which Lorentz frame does the objective collapse happen in? Another alternative is to do without collapse of any kind subjective or objective. What is important, then, are the EPR correlations between what is measured and the measuring apparatus. These correlations are the effects of the physical interaction and the arrow of time is built in unless one also permits future causes of past correlations. Aharonov has a post-quantum theory in which present correlation have both past and future causes. Does Smolin's self-organizing emerging universe permit future causes in which what has not yet actually happened influences what has happened and what is happening? This is more than just metaphysics, this bears on the problem of whether destiny and design are mere delusions or something physically real subject to decidable Popperian falsification. I think that is the case.
The Everett many-worlds permits superpositions of the observer unless additional ad hoc assumptions are made. John Bell discusses this in Speakable and unspeakable in quantum mechanics. One way to explain why we do not experience macroscopic superpositions is with environmental decoherence as in the "consistent classical histories" of Gell-Mann and Hartle. Smolin proceeds with a projection postulate as in the theory of Bohr. Smolin disagrees with Gell-Mann, for example, that one must renounce Bohr in order to do quantum cosmology. (p. 30) All our measurements are incomplete so there is no problem that we are inside the universe as long as we refrain from making measurements on our own brains. Smolin seems to have changed his mind on this later on in his popular book The Life of the Cosmos -- but I could be mistaken here.
Smolin thinks it is important that the areas in quantum gravity are discrete and that this is a robust result independent of model details. Smolin shows how his theory leads to the semiclassical theory when gravity degrees of freedom are classical. Strangely, he drops the requirement of diffeomorphism invariance in doing this. (p. 33) He uses a "weave state"-- that's "weave" not "wave". What is interesting is that his model using a single matter scalar field seems to give a 1+1 string theory in the weak semiclassical limit of a fixed FLAT spacetime metric. A large zero point energy is required to get the correct limit. This Lorentz vacuum state is a Glauber coherent state i.e., a minimum uncertainty wavepacket in the 3-metric and its conjugate from the weave state. (p. 35) This Lorentz vacuum consists mostly of large loops corresponding to a Gaussian distribution of virtual gravitons. Back-reaction and coupling to the gravitons are neglected here. There is no mass and self interaction in the scalar matter field of Smolin's toy model here. To add mass and self-interaction matter terms requires adding volume operators to weave state construction. This causes back-reaction of the matter field on the metric to construct solutions of the Hamiltonian constraint. (p. 36) The special unitary inner product of field theory need not be recovered from the full quantum gravity inner product if part of the global time clock field becomes engulfed in a gravitatonal collapse. (p 44) Is the apparent loss of information and quantum coherence down the semiclassical black hole there in the full quantum gravity model? This requires a clear notion of global clock time such as Smolin has attempted. If quantum gravity eliminates the classical Penrose-Hawking metric singularities then then information can be recovered and the decoherence can be quantum erased. This is an open question as the singularities may survive. If the singularities survive then unitarity is violated and many physicists do not like that. I think unitarity violation, i.e., nonconservation of total probability, is required for the introduction of novelty in the world. Smolin says that global time is not absolute but depends upon contingencies. (p. 37) Smolin argues that diffeomorphism invariance, with its consequent gravitational collapse, is fundamentally antithetical to unitarity which demands a Newtonian absolute time. (p. 38) Smolin does use the positive energy condition which excludes the "exotic matter" required for, at least some kinds of, traversable wormholes and warp drives. It is a dynamical question, not a matter of principle, on whether the classical singularities survive in a full quantum gravity theory. (p. 40) Smolin then asks if there is an alternative not based on his operational clock time field? If the Hilbert space structure is contingent then, in the Bohr paradigm, not only the actualities, but the potentialities depend contingently and subjectively on the physical frames. Is an objective frame-independent set of potentialities possible? Smolin describes ideas of Barbour that may restore the objectivity of quantum potentialities where the quantum gravity Hilbert space inner product is not frame, or observer, dependent. Barbour says time does not actually exist. I think Barbour has borrowed an idea that Fred Hoyle used in his science fiction novel, The Black Cloud. There was also a comic book in the 40's called "The Heap". The idea is that our experience of the passage of time, our getting older etc, is an illusion caused by certain properties of the classical limit of quantum cosmology. Barbour's "heap hypothesis" that "The world consists of a timeless real ensemble of configurations called 'the heap'." The "configurations" are global "moments". The probability for any given spatially diffiomorphism invariant observable property to occur in the heap is determined by an objective quantum gravity wavefunction of the universe. This probability is an "actual ensemble average". (p.45) Barbour's inner product is spatially diffeomorphic invariant including any possible clock fields. This is different from Smolin's previous construction where the clocks were not included in the spatial diffeomorphism invariance, but were "external". There is no "time" in Barbour's theory. Our memories are part of the wavefunction of the universe. Each moment has a complete set of memories different from one moment to another. Causal connections between different moments are only contingent artifacts of the semiclassical limit feature of the wavefunction of the universe. However, Smolin points to an ambiguity on how we, who are inside the universe, are to interpret Barbour's "probability". If probability is defined to us inside the universe, there is no apriori reason to clutch on to unitarity and conservation of probability. (p.45) This is one of Smolin's basic conclusions that I agree with on other grounds.