Microcausality

If e0 ∈ R1+1 is a future-directed timelike unit vector, and if e1 is the unique spacelike unit vector with e0e1 = 0 that “points to the right,” then coordinates x0 and x1 on R1+1 are defined by x0(q) := qe0 and x1(q) := qe1. The partial differential operator

x : = ∂2x0 − ∂2x1

does not depend on the choice of e0.

The Fourier transform of the Klein-Gordon equation

(□ + m2)u = 0 —– (1)

where m > 0 is a given mass, is

(−p2 + m2)û(p) = 0 —– (2)

As a consequence, the support of û has to be a subset of the hyperbola Hm ⊂ R1+1 specified by the condition p2 = m2. One connected component of Hm consists of positive-energy vectors only; it is called the upper mass shell Hm+. The elements of Hm+ are the 4-momenta of classical relativistic point particles.

Denote by L1 the restricted Lorentz group, i.e., the connected component of the Lorentz group containing its unit element. In 1 + 1 dimensions, L1 coincides with the one-parameter Abelian group B(χ), χ ∈ R, of boosts. Hm+ is an orbit of L1 without fixed points. So if one chooses any point p′ ∈ Hm+, then there is, for each p ∈ Hm+, a unique χ(p) ∈ R with p = B(χ(p))p′. By construction, χ(B(ξ)p) = χ(p) + ξ, so the measure dχ on Hm+ is invariant under boosts and does note depend on the choice of p′.

For each p ∈ Hm+, the plane wave q ↦ e±ipq on R1+1 is a classical solution of the Klein-Gordon equation. The Klein-Gordon equation is linear, so if a+ and a are, say, integrable functions on Hm+, then

F(q) := ∫Hm+ (a+(p)e-ipq + a(p)eipq dχ(p) —– (3)

is a solution of the Klein-Gordon equation as well. If the functions a± are not integrable, the field F may still be well defined as a distribution. As an example, put a± ≡ (2π)−1, then

F(q) = (2π)−1 Hm+ (e-ipq + eipq) dχ(p) = π−1Hm+ cos(pq) dχ(p) =: Φ(q) —– (4)

and for a± ≡ ±(2πi)−1, F equals

F(q) = (2πi)−1Hm+ (e-ipq – eipq) dχ(p) = π−1Hm+ sin(pq) dχ(p) =: ∆(q) —– (5)

Quantum fields are obtained by “plugging” classical field equations and their solutions into the well-known second quantization procedure. This procedure replaces the complex (or, more generally speaking, finite-dimensional vector) field values by linear operators in an infinite-dimensional Hilbert space, namely, a Fock space. The Hilbert space of the hermitian scalar field is constructed from wave functions that are considered as the wave functions of one or several particles of mass m. The single-particle wave functions are the elements of the Hilbert space H1 := L2(Hm+, dχ). Put the vacuum (zero-particle) space H0 equal to C, define the vacuum vector Ω := 1 ∈ H0, and define the N-particle space HN as the Hilbert space of symmetric wave functions in L2((Hm+)N, dNχ), i.e., all wave functions ψ with

ψ(pπ(1) ···pπ(N)) = ψ(p1 ···pN)

∀ permutations π ∈ SN. The bosonic Fock space H is defined by

H := ⊕N∈N HN.

The subspace

D := ∪M∈N ⊕0≤M≤N HN is called a finite particle space.

The definition of the N-particle wave functions as symmetric functions endows the field with a Bose–Einstein statistics. To each wave function φ ∈ H1, assign a creation operator a+(φ) by

a+(φ)ψ := CNφ ⊗s ψ, ψ ∈ D,

where ⊗s denotes the symmetrized tensor product and where CN is a constant.

(a+(φ)ψ)(p1 ···pN) = CN/N ∑v φ(pν)ψ(pπ(1) ···p̂ν ···pπ(N)) —– (6)

where the hat symbol indicates omission of the argument. This defines a+(φ) as a linear operator on the finite-particle space D.

The adjoint operator a(φ) := a+(φ) is called an annihilation operator; it assigns to each ψ ∈ HN, N ≥ 1, the wave function a(φ)ψ ∈ HN−1 defined by

(a(φ)ψ)(p1 ···pN) := CN ∫Hm+ φ(p)ψ(p1 ···pN−1, p) dχ(p)

together with a(φ)Ω := 0, this suffices to specify a(φ) on D. Annihilation operators can also be defined for sharp momenta. Namely, one can define to each p ∈ Hm+ the annihilation operator a(p) assigning to

each ψ ∈ HN, N ≥ 1, the wave function a(p)ψ ∈ HN−1 given by

(a(p)ψ)(p1 ···pN−1) := Cψ(p, p1 ···pN−1), ψ ∈ HN,

and assigning 0 ∈ H to Ω. a(p) is, like a(φ), well defined on the finite-particle space D as an operator, but its hermitian adjoint is ill-defined as an operator, since the symmetric tensor product of a wave function by a delta function is no wave function.

Given any single-particle wave functions ψ, φ ∈ H1, the commutators [a(ψ), a(φ)] and [a+(ψ), a+(φ)] vanish by construction. It is customary to choose the constants CN in such a fashion that creation and annihilation operators exhibit the commutation relation

[a(φ), a+(ψ)] = ⟨φ, ψ⟩ —– (7)

which requires CN = N. With this choice, all creation and annihilation operators are unbounded, i.e., they are not continuous.

When defining the hermitian scalar field as an operator valued distribution, it must be taken into account that an annihilation operator a(φ) depends on its argument φ in an antilinear fashion. The dependence is, however, R-linear, and one can define the scalar field as a C-linear distribution in two steps.

For each real-valued test function φ on R1+1, define

Φ(φ) := a(φˆ|Hm+) + a+(φˆ|Hm+)

then one can define for an arbitrary complex-valued φ

Φ(φ) := Φ(Re(φ)) + iΦ(Im(φ))

Referring to (4), Φ is called the hermitian scalar field of mass m.

Thereafter, one could see

[Φ(q), Φ(q′)] = i∆(q − q′) —– (8)

Referring to (5), which is to be read as an equation of distributions. The distribution ∆ vanishes outside the light cone, i.e., ∆(q) = 0 if q2 < 0. Namely, the integrand in (5) is odd with respect to some p′ ∈ Hm+ if q is spacelike. Note that pq > 0 for all p ∈ Hm+ if q ∈ V+. The consequence of this is called microcausality: field operators located in spacelike separated regions commute (for the hermitian scalar field).

Revisiting Catastrophes. Thought of the Day 134.0

The most explicit influence from mathematics in semiotics is probably René Thom’s controversial theory of catastrophes (here and here), with philosophical and semiotic support from Jean Petitot. Catastrophe theory is but one of several formalisms in the broad field of qualitative dynamics (comprising also chaos theory, complexity theory, self-organized criticality, etc.). In all these cases, the theories in question are in a certain sense phenomenological because the focus is different types of qualitative behavior of dynamic systems grasped on a purely formal level bracketing their causal determination on the deeper level. A widespread tool in these disciplines is phase space – a space defined by the variables governing the development of the system so that this development may be mapped as a trajectory through phase space, each point on the trajectory mapping one global state of the system. This space may be inhabited by different types of attractors (attracting trajectories), repellors (repelling them), attractor basins around attractors, and borders between such basins characterized by different types of topological saddles which may have a complicated topology.

Catastrophe theory has its basis in differential topology, that is, the branch of topology keeping various differential properties in a function invariant under transformation. It is, more specifically, the so-called Whitney topology whose invariants are points where the nth derivative of a function takes the value 0, graphically corresponding to minima, maxima, turning tangents, and, in higher dimensions, different complicated saddles. Catastrophe theory takes its point of departure in singularity theory whose object is the shift between types of such functions. It thus erects a distinction between an inner space – where the function varies – and an outer space of control variables charting the variation of that function including where it changes type – where, e.g. it goes from having one minimum to having two minima, via a singular case with turning tangent. The continuous variation of control parameters thus corresponds to a continuous variation within one subtype of the function, until it reaches a singular point where it discontinuously, ‘catastrophically’, changes subtype. The philosophy-of-science interpretation of this formalism now conceives the stable subtype of function as representing the stable state of a system, and the passage of the critical point as the sudden shift to a new stable state. The configuration of control parameters thus provides a sort of map of the shift between continuous development and discontinuous ‘jump’. Thom’s semiotic interpretation of this formalism entails that typical catastrophic trajectories of this kind may be interpreted as stable process types phenomenologically salient for perception and giving rise to basic verbal categories.

One of the simpler catastrophes is the so-called cusp (a). It constitutes a meta-diagram, namely a diagram of the possible type-shifts of a simpler diagram (b), that of the equation ax4 + bx2 + cx = 0. The upper part of (a) shows the so-called fold, charting the manifold of solutions to the equation in the three dimensions a, b and c. By the projection of the fold on the a, b-plane, the pointed figure of the cusp (lower a) is obtained. The cusp now charts the type-shift of the function: Inside the cusp, the function has two minima, outside it only one minimum. Different paths through the cusp thus corresponds to different variations of the equation by the variation of the external variables a and b. One such typical path is the path indicated by the left-right arrow on all four diagrams which crosses the cusp from inside out, giving rise to a diagram of the further level (c) – depending on the interpretation of the minima as simultaneous states. Here, thus, we find diagram transformations on three different, nested levels.

The concept of transformation plays several roles in this formalism. The most spectacular one refers, of course, to the change in external control variables, determining a trajectory through phase space where the function controlled changes type. This transformation thus searches the possibility for a change of the subtypes of the function in question, that is, it plays the role of eidetic variation mapping how the function is ‘unfolded’ (the basic theorem of catastrophe theory refers to such unfolding of simple functions). Another transformation finds stable classes of such local trajectory pieces including such shifts – making possible the recognition of such types of shifts in different empirical phenomena. On the most empirical level, finally, one running of such a trajectory piece provides, in itself, a transformation of one state into another, whereby the two states are rationally interconnected. Generally, it is possible to make a given transformation the object of a higher order transformation which by abstraction may investigate aspects of the lower one’s type and conditions. Thus, the central unfolding of a function germ in Catastrophe Theory constitutes a transformation having the character of an eidetic variation making clear which possibilities lie in the function germ in question. As an abstract formalism, the higher of these transformations may determine the lower one as invariant in a series of empirical cases.

Complexity theory is a broader and more inclusive term covering the general study of the macro-behavior of composite systems, also using phase space representation. The theoretical biologist Stuart Kauffman (intro) argues that in a phase space of all possible genotypes, biological evolution must unfold in a rather small and specifically qualified sub-space characterized by many, closely located and stable states (corresponding to the possibility of a species to ‘jump’ to another and better genotype in the face of environmental change) – as opposed to phase space areas with few, very stable states (which will only be optimal in certain, very stable environments and thus fragile when exposed to change), and also opposed, on the other hand, to sub-spaces with a high plurality of only metastable states (here, the species will tend to merge into neighboring species and hence never stabilize). On the base of this argument, only a small subset of the set of virtual genotypes possesses ‘evolvability’ as this special combination between plasticity and stability. The overall argument thus goes that order in biology is not a pure product of evolution; the possibility of order must be present in certain types of organized matter before selection begins – conversely, selection requires already organized material on which to work. The identification of a species with a co-localized group of stable states in genome space thus provides a (local) invariance for the transformation taking a trajectory through space, and larger groups of neighboring stabilities – lineages – again provide invariants defined by various more or less general transformations. Species, in this view, are in a certain limited sense ‘natural kinds’ and thus naturally signifying entities. Kauffman’s speculations over genotypical phase space have a crucial bearing on a transformation concept central to biology, namely mutation. On this basis far from all virtual mutations are really possible – even apart from their degree of environmental relevance. A mutation into a stable but remotely placed species in phase space will be impossible (evolution cannot cross the distance in phase space), just like a mutation in an area with many, unstable proto-species will not allow for any stabilization of species at all and will thus fall prey to arbitrary small environment variations. Kauffman takes a spontaneous and non-formalized transformation concept (mutation) and attempts a formalization by investigating its condition of possibility as movement between stable genomes in genotype phase space. A series of constraints turn out to determine type formation on a higher level (the three different types of local geography in phase space). If the trajectory of mutations must obey the possibility of walking between stable species, then the space of possibility of trajectories is highly limited. Self-organized criticality as developed by Per Bak (How Nature Works the science of self-organized criticality) belongs to the same type of theories. Criticality is here defined as that state of a complicated system where sudden developments in all sizes spontaneously occur.

Nomological Unification and Phenomenology of Gravitation. Thought of the Day 110.0

String theory, which promises to give an all-encompassing, nomologically unified description of all interactions did not even lead to any unambiguous solutions to the multitude of explanative desiderata of the standard model of quantum field theory: the determination of its specific gauge invariances, broken symmetries and particle generations as well as its 20 or more free parameters, the chirality of matter particles, etc. String theory does at least give an explanation for the existence and for the number of particle generations. The latter is determined by the topology of the compactified additional spatial dimensions of string theory; their topology determines the structure of the possible oscillation spectra. The number of particle generations is identical to half the absolute value of the Euler number of the compact Calabi-Yau topology. But, because it is completely unclear which topology should be assumed for the compact space, there are no definitive results. This ambiguity is part of the vacuum selection problem; there are probably more than 10100 alternative scenarios in the so-called string landscape. Moreover all concrete models, deliberately chosen and analyzed, lead to generation numbers much too big. There are phenomenological indications that the number of particle generations can not exceed three. String theory admits generation numbers between three and 480.

Attempts at a concrete solution of the relevant external problems (and explanative desiderata) either did not take place, or they did not show any results, or they led to escalating ambiguities and finally got drowned completely in the string landscape scenario: the recently developed insight that string theory obviously does not lead to a unique description of nature, but describes an immense number of nomologically, physically and phenomenologically different worlds with different symmetries, parameter values, and values of the cosmological constant.

String theory seems to be by far too much preoccupied with its internal conceptual and mathematical problems to be able to find concrete solutions to the relevant external physical problems. It is almost completely dominated by internal consistency constraints. It is not the fact that we are living in a ten-dimensional world which forces string theory to a ten-dimensional description. It is that perturbative string theories are only anomaly-free in ten dimensions; and they contain gravitons only in a ten-dimensional formulation. The resulting question, how the four-dimensional spacetime of phenomenology comes off from ten-dimensional perturbative string theories (or its eleven-dimensional non-perturbative extension: the mysterious, not yet existing M theory), led to the compactification idea and to the braneworld scenarios, and from there to further internal problems.

It is not the fact that empirical indications for supersymmetry were found, that forces consistent string theories to include supersymmetry. Without supersymmetry, string theory has no fermions and no chirality, but there are tachyons which make the vacuum instable; and supersymmetry has certain conceptual advantages: it leads very probably to the finiteness of the perturbation series, thereby avoiding the problem of non-renormalizability which haunted all former attempts at a quantization of gravity; and there is a close relation between supersymmetry and Poincaré invariance which seems reasonable for quantum gravity. But it is clear that not all conceptual advantages are necessarily part of nature, as the example of the elegant, but unsuccessful Grand Unified Theories demonstrates.

Apart from its ten (or eleven) dimensions and the inclusion of supersymmetry, both have more or less the character of only conceptually, but not empirically motivated ad-hoc assumptions. String theory consists of a rather careful adaptation of the mathematical and model-theoretical apparatus of perturbative quantum field theory to the quantized, one-dimensionally extended, oscillating string (and, finally, of a minimal extension of its methods into the non-perturbative regime for which the declarations of intent exceed by far the conceptual successes). Without any empirical data transcending the context of our established theories, there remains for string theory only the minimal conceptual integration of basic parts of the phenomenology already reproduced by these established theories. And a significant component of this phenomenology, namely the phenomenology of gravitation, was already used up in the selection of string theory as an interesting approach to quantum gravity. Only, because string theory, containing gravitons as string states, reproduces in a certain way the phenomenology of gravitation, it is taken seriously.

Weyl and Automorphism of Nature. Drunken Risibility.

In classical geometry and physics, physical automorphisms could be based on the material operations used for defining the elementary equivalence concept of congruence (“equality and similitude”). But Weyl started even more generally, with Leibniz’ explanation of the similarity of two objects, two things are similar if they are indiscernible when each is considered by itself. Here, like at other places, Weyl endorsed this Leibnzian argument from the point of view of “modern physics”, while adding that for Leibniz this spoke in favour of the unsubstantiality and phenomenality of space and time. On the other hand, for “real substances” the Leibnizian monads, indiscernability implied identity. In this way Weyl indicated, prior to any more technical consideration, that similarity in the Leibnizian sense was the same as objective equality. He did not enter deeper into the metaphysical discussion but insisted that the issue “is of philosophical significance far beyond its purely geometric aspect”.

Weyl did not claim that this idea solves the epistemological problem of objectivity once and for all, but at least it offers an adequate mathematical instrument for the formulation of it. He illustrated the idea in a first step by explaining the automorphisms of Euclidean geometry as the structure preserving bijective mappings of the point set underlying a structure satisfying the axioms of “Hilbert’s classical book on the Foundations of Geometry”. He concluded that for Euclidean geometry these are the similarities, not the congruences as one might expect at a first glance. In the mathematical sense, we then “come to interpret objectivity as the invariance under the group of automorphisms”. But Weyl warned to identify mathematical objectivity with that of natural science, because once we deal with real space “neither the axioms nor the basic relations are given”. As the latter are extremely difficult to discern, Weyl proposed to turn the tables and to take the group Γ of automorphisms, rather than the ‘basic relations’ and the corresponding relata, as the epistemic starting point.

Hence we come much nearer to the actual state of affairs if we start with the group Γ of automorphisms and refrain from making the artificial logical distinction between basic and derived relations. Once the group is known, we know what it means to say of a relation that it is objective, namely invariant with respect to Γ.

By such a well chosen constitutive stipulation it becomes clear what objective statements are, although this can be achieved only at the price that “…we start, as Dante starts in his Divina Comedia, in mezzo del camin”. A phrase characteristic for Weyl’s later view follows:

It is the common fate of man and his science that we do not begin at the beginning; we find ourselves somewhere on a road the origin and end of which are shrouded in fog.

Weyl’s juxtaposition of the mathematical and the physical concept of objectivity is worthwhile to reflect upon. The mathematical objectivity considered by him is relatively easy to obtain by combining the axiomatic characterization of a mathematical theory with the epistemic postulate of invariance under a group of automorphisms. Both are constituted in a series of acts characterized by Weyl as symbolic construction, which is free in several regards. For example, the group of automorphisms of Euclidean geometry may be expanded by “the mathematician” in rather wide ways (affine, projective, or even “any group of transformations”). In each case a specific realm of mathematical objectivity is constituted. With the example of the automorphism group Γ of (plane) Euclidean geometry in mind Weyl explained how, through the use of Cartesian coordinates, the automorphisms of Euclidean geometry can be represented by linear transformations “in terms of reproducible numerical symbols”.

For natural science the situation is quite different; here the freedom of the constitutive act is severely restricted. Weyl described the constraint for the choice of Γ at the outset in very general terms: The physicist will question Nature to reveal him her true group of automorphisms. Different to what a philosopher might expect, Weyl did not mention, the subtle influences induced by theoretical evaluations of empirical insights on the constitutive choice of the group of automorphisms for a physical theory. He even did not restrict the consideration to the range of a physical theory but aimed at Nature as a whole. Still basing on his his own views and radical changes in the fundamental views of theoretical physics, Weyl hoped for an insight into the true group of automorphisms of Nature without any further specifications.

Weyl’s Lagrange Density of General Relativistic Maxwell Theory

Weyl pondered on the reasons why the structure group of the physical automorphisms still contained the “Euclidean rotation group” (respectively the Lorentz group) in such a prominent role:

The Euclidean group of rotations has survived even such radical changes of our concepts of the physical world as general relativity and quantum theory. What then are the peculiar merits of this group to which it owes its elevation to the basic group pattern of the universe? For what ‘sufficient reasons’ did the Creator choose this group and no other?”

He reminded that Helmholtz had characterized ∆o ≅ SO (3, ℜ) by the “fact that it gives to a rotating solid what we may call its just degrees of freedom” of a rotating solid body; but this method “breaks down for the Lorentz group that in the four-dimensional world takes the place of the orthogonal group in 3-space”. In the early 1920s he himself had given another characterization living up to the new demands of the theories of relativity in his mathematical analysis of the problem of space.

He mentioned the idea that the Lorentz group might play its prominent role for the physical automorphisms because it expresses deep lying matter structures; but he strongly qualified the idea immediately after having stated it:

Since we have the dualism of invariance with respect to two groups and Ω certainly refers to the manifold of space points, it is a tempting idea to ascribe ∆o to matter and see in it a characteristic of the localizable elementary particles of matter. I leave it undecided whether this idea, the very meaning of which is somewhat vague, has any real merits.

. . . But instead of analysing the structure of the orthogonal group of transformations ∆o, it may be wiser to look for a characterization of the group ∆o as an abstract group. Here we know that the homogeneous n-dimensional orthogonal groups form one of 3 great classes of simple Lie groups. This is at least a partial solution of the problem.

He left it open why it ought to be “wiser” to look for abstract structure properties in order to answer a natural philosophical question. Could it be that he wanted to indicate an open-mindedness toward the more structuralist perspective on automorphism groups, preferred by the young algebraists around him at Princetion in the 1930/40s? Today the classification of simple Lie groups distinguishes 4 series, Ak,Bk,Ck,Dk. Weyl apparently counted the two orthogonal series Bk and Dk as one. The special orthogonal groups in even complex space dimension form the series of simple Lie groups of type Dk, with complex form (SO 2k,C) and real compact form (SO 2k,ℜ). The special orthogonal group in odd space dimension form the series type Bk, with complex form SO(2k + 1, C) and compact real form SO(2k + 1, ℜ).

But even if one accepted such a general structuralist view as a starting point there remained a question for the specification of the space dimension of the group inside the series.

But the number of the dimensions of the world is 4 and not an indeterminate n. It is a fact that the structure of ∆o is quite different for the various dimensionalities n. Hence the group may serve as a clue by which to discover some cogent reason for the di- mensionality 4 of the world. What must be brought to light, is the distinctive character of one definite group, the four-dimensional Lorentz group, either as a group of linear transformations, or as an abstract group.

The remark that the “structure of ∆o is quite different for the various dimensionalities n” with regard to even or odd complex space dimensions (type Dk, resp. Bk) strongly qualifies the import of the general structuralist characterization. But already in the 1920s Weyl had used the fact that for the (real) space dimension n “4 the universal covering of the unity component of the Lorentz group SO (1, 3)o is the realification of SL (2, C). The latter belongs to the first of the Ak series (with complex form SL (k + 1,C). Because of the isomorphism of the initial terms of the series, A1 ≅ B1, this does not imply an exception of Weyl’s general statement. We even may tend to interpret Weyl’s otherwise cryptic remark that the structuralist perspective gives a “at least a partial solution of the problem” by the observation that the Lorentz group in dimension n “4 is, in a rather specific way, the realification of the complex form of one of the three most elementary non-commutative simple Lie groups of type A1 ≅ B1. Its compact real form is SO (3, ℜ), respectively the latter’s universal cover SU (2, C).

Weyl stated clearly that the answer cannot be expected by structural considerations alone. The problem is only “partly one of pure mathematics”, the other part is “empirical”. But the question itself appeared of utmost importance to him

We can not claim to have understood Nature unless we can establish the uniqueness of the four-dimensional Lorentz group in this sense. It is a fact that many of the known laws of nature can at once be generalized to n dimensions. We must dig deep enough until we hit a layer where this is no longer the case.

In 1918 he had given an argument why, in the framework of his new scale gauge geometry, the “world” had to be of dimension 4. His argument had used the construction of the Lagrange density of general relativistic Maxwell theory Lf = fμν fμν √(|detg|), with fμν the components of curvature of his newly introduced scale/length connection, physically interpreted by him as the electromagnetic field. Lf is scale invariant only in spacetime dimension n = 4. The shift from scale gauge to phase gauge undermined the importance of this argument. Although it remained correct mathematically, it lost its convincing power once the scale gauge transformations were relegated from physics to the mathematical automorphism group of the theory only.

Weyl said:

Our question has this in common with most questions of philosophical nature: it depends on the vague distinction between essential and non-essential. Several competing solutions are thinkable; but it may also happen that, once a good solution of the problem is found, it will be of such cogency as to command general recognition.

Diffeomorphism Invariance: General Relativity Spacetime Points Cannot Possess Haecceity.

Eliminative or radical ontic structural realism (ROSR) offers a radical cure—appropriate given its name—to what it perceives to be the ailing of traditional, object-based realist interpretations of fundamental theories in physics: rid their ontologies entirely of objects. The world does not, according to this view, consist of fundamental objects, which may or may not be individuals with a well-defined intrinsic identity, but instead of physical structures that are purely relational in the sense of networks of ‘free-standing’ physical relations without relata.

Advocates of ROSR have taken at least three distinct issues in fundamental physics to support their case. The quantum statistical features of an ensemble of elementary quantum particles of the same kind as well as the features of entangled elementary quantum (field) systems as illustrated in the violation of Bell-type inequalities challenge the standard understanding of the identity and individuality of fundamental physical objects: considered on their own, an elementary quantum particle part of the above mentioned ensemble or an entangled elementary quantum system (that is, an elementary quantum system standing in a quantum entanglement relation) cannot be said to satisfy genuine and empirically meaningful identity conditions. Thirdly, it has been argued that one of the consequences of the diffeomorphism invariance and background independence found in general relativity (GTR) is that spacetime points should not be considered as traditional objects possessing some haecceity, i.e. some identity on their own.

The trouble with ROSR is that its main assertion appears squarely incoherent: insofar as relations can be exemplified, they can only be exemplified by some relata. Given this conceptual dependence of relations upon relata, any contention that relations can exist floating freely from some objects that stand in those relations seems incoherent. If we accept an ontological commitment e.g. to universals, we may well be able to affirm that relations exist independently of relata – as abstracta in a Platonic heaven. The trouble is that ROSR is supposed to be a form of scientific realism, and as such committed to asserting that at least certain elements of the relevant theories of fundamental physics faithfully capture elements of physical reality. Thus, a defender of ROSR must claim that, fundamentally, relations-sans-relata are exemplified in the physical world, and that contravenes both the intuitive and the usual technical conceptualization of relations.

The usual extensional understanding of n-ary relations just equates them with subsets of the n-fold Cartesian product of the set of elementary objects assumed to figure in the relevant ontology over which the relation is defined. This extensional, ultimately set-theoretic, conceptualization of relations pervades philosophy and operates in the background of fundamental physical theories as they are usually formulated, as well as their philosophical appraisal in the structuralist literature. The charge then is that the fundamental physical structures that are represented in the fundamental physical theories are just not of the ‘object-free’ type suggested by ROSR.

While ROSR should not be held to the conceptual standards dictated by the metaphysical prejudices it denies, giving up the set-theoretical framework and the ineliminable reference to objects and relata attending its characterizations of relations and structure requires an alternative conceptualization of these notions so central to the position. This alternative conceptualization remains necessary even in the light of ‘metaphysics first’ complaints, which insist that ROSR’s problem must be confronted, first and foremost, at the metaphysical level, and that the question of how to represent structure in our language and in our theories only arises in the wake of a coherent metaphysical solution. But the radical may do as much metaphysics as she likes, articulate her theory and her realist commitments she must, and in order to do that, a coherent conceptualization of what it is to have free-floating relations exemplified in the physical world is necessary.

ROSR thus confronts a dilemma: either soften to a more moderate structural realist position or else develop the requisite alternative conceptualizations of relations and of structures and apply them to fundamental physical theories. A number of structural realists have grabbed the first leg and proposed less radical and non-eliminative versions of ontic structural realism (OSR). These moderate cousins of ROSR aim to take seriously the difficulties of the traditional metaphysics of objects for understanding fundamental physics while avoiding these major objections against ROSR by keeping some thin notion of object. The picture typically offered is that of a balance between relations and their relata, coupled to an insistence that these relata do not possess their identity intrinsically, but only by virtue of occupying a relational position in a structural complex. Because it strikes this ontological balance, we term this moderate version of OSR ‘balanced ontic structural realism’ (BOSR).

But holding their ground may reward the ROSRer with certain advantages over its moderate competitors. First, were the complete elimination of relata to succeed, then structural realism would not confront any of the known headaches concerning the identity of these objects or, relatedly, the status of the Principle of the Identity of Indiscernibles. To be sure, this embarrassment can arguably be avoided by other moves; but eliminating objects altogether simply obliterates any concerns whether two objects are one and the same. Secondly, and speculatively, alternative formulations of our fundamental physical theories may shed light on a path toward a quantum theory of gravity.

For these presumed advantages to come to bear, however, the possibility of a precise formulation of the notion of ‘free-standing’ (or ‘object-free’) structure, in the sense of a network of relations without relata (without objects) must thus be achieved.  Jonathan Bain has argued that category theory provides the appropriate mathematical framework for ROSR, allowing for an ‘object-free’ notion of relation, and hence of structure. This argument can only succeed, however, if the category-theoretical formulation of (some of the) fundamental physical theories has some physical salience that the set-theoretical formulation lacks, or proves to be preferable qua formulation of a physical theory in some other way.

F. A. Muller has argued that neither set theory nor category theory provide the tools necessary to clarify the “Central Claim” of structural realism that the world, or parts of the world, have or are some structure. The main reason for this arises from the failure of reference in the contexts of both set theory and category theory, at least if some minimal realist constraints are imposed on how reference can function. Consequently, Muller argues that an appropriately realist stucturalist is better served by fixing the concept of structure by axiomatization rather than by (set-theoretical or category-theoretical) definition.

Are Categories Similar to Sets? A Folly, if General Relativity and Quantum Mechanics Think So.

The fundamental importance of the path integral suggests that it might be enlightening to simplify things somewhat by stripping away the knot observable K and studying only the bare partition functions of the theory, considered over arbitrary spacetimes. That is, consider the path integral

Z(M) = ∫ DA e (i ∫M S(A) —– (1)

where M is an arbitrary closed 3d manifold, that is, compact and without boundary, and S[A] is the Chern-Simons action. Immediately one is struck by the fact that, since the action is topological, the number Z(M) associated to M should be a topological invariant of M. This is a remarkably efficient way to produce topological invariants.

Poincaré Conjecture: If M is a closed 3-manifold, whose fundamental group π1(M), and all of whose homology groups Hi(M) are equal to those of S3, then M is homeomorphic to S3.

One therefore appreciates the simplicity of the quantum field theory approach to topological invariants, which runs as follows.

1. Endow the space with extra geometric structure in the form of a connection (alternatively a field, a section of a line bundle, an embedding map into spacetime)
2. Compute a number from this manifold-with-connection (the action)
3. Sum over all connections.

This may be viewed as an extension of the general principle in mathematics that one should classify structures by the various kinds of extra structure that can live on them. Indeed, the Chern-Simons Lagrangian was originally introduced in mathematics in precisely this way. Chern-Weil theory provides access to the cohomology groups (that is, topological invariants) of a manifold M by introducing an arbitrary connection A on M, and then associating to A a closed form f(A) (for instance, via the Chern-Simons Lagrangian), whose cohomology class is, remarkably, independent of the original arbitrary choice of connection A. Quantum field theory takes this approach to the extreme by being far more ambitious; it associates to a connection A the actual numerical value of the action (usually obtained by integration over M) – this number certainly depends on the connection, but field theory atones for this by summing over all connections.

Quantum field theory is however, in its path integral manifestation, far more than a mere machine for computing numbers associated with manifolds. There is dynamics involved, for the natural purpose of path integrals is not to calculate bare partition functions such as equation (1), but rather to express the probability amplitude for a given field configuration to evolve into another. Thus one considers a 3d manifold M (spacetime) with boundary components Σ1 and Σ2 (space), and considers M as the evolution of space from its initial configuration Σ1 to its final configuration Σ2:

This is known mathematically as a cobordism from Σ1 to Σ2. To a 2d closed manifold Σ we associate the space of fields A(Σ) living on Σ. A physical state Ψ corresponds to a functional on this space of fields. This is the Schrödinger picture of quantum field theory: if A ∈ A(Σ), then Ψ(A) represents the probability that the state known as Ψ will be found in the field A. Such a state evolves with time due to the dynamics of the theory; Ψ(A) → Ψ(A, t). The space of states has a natural basis, which consists of the delta functionals Â – these are the states satisfying ⟨Â|Â′⟩ = δ(A − A′). Any arbitrary state Ψ may be expressed as a superposition of these basis states. The path integral instructs us how to compute the time evolution of states, by first expanding them in the Â basis, and then specifying that the amplitude for a system in the state Â1 on the space Σ1 to be found in the state Â2 on the space Σ2 is given by:

〈Â2|U|Â1〉= ∫A | ∑2 = A2 A | ∑1 = A1 DA e i S[A] —– (2)

This equation is the fundamental formula of quantum field theory: ‘Perform a weighted sum over all possible fields (connections) living on spacetime that restrict to A1 and A2 on Σ1 and Σ2 respectively’. This formula constructs the time evolution operator U associated to the cobordism M.

In this way we see that, at the very heart of quantum mechanics and quantum field theory, is a formula which associates to every space-like manifold Σ a Hilbert space of fields A(Σ), and to every cobordism M from Σ1 to Σ2 a time evolution operator U(M) : Σ1 – Σ2. To specify a quantum field theory is nothing more than to give rules for constructing the Hilbert spaces A(Σ) and the rules (correlation functions) for calculating the time evolution operators U(M). This is precisely the statement that a quantum field theory is a functor from the cobordism category nCob to the category of Hilbert spaces Hilb.

A category C consists of a collection of objects, a collection of arrows f:a → b from any object a to any object b, a rule for composing arrows f:a → b and g : b → c to obtain an arrow g f : a → c, and for each object A an identity arrow 1a : a → a. These must satisfy the associative law f(gh) = (fg)h and the left and right unit laws 1af = f and f1a = f whenever these composites are defined. In many cases, the objects of a category are best thought of as sets equipped with extra structure, while the morphisms are functions preserving the structure. However, this is neither true for the category of Hilbert spaces nor for the category of cobordisms.

The fundamental idea of category theory is to consider the ‘external’ structure of the arrows between objects instead of the ‘internal’ structure of the objects themselves – that is, the actual elements inside an object – if indeed, an object is a set at all : it need not be, since category theory waives its right to ask questions about what is inside an object, but reserves its right to ask how one object is related to another.

A functor F : C → D from a category C to another category D is a rule which associates to each object a of C an object b of D, and to each arrow f :a → b in C a corresponding arrow F(f): F(a) → F(b) in D. This association must preserve composition and the units, that is, F(fg) = F(f)F(g) and F(1a) = 1F(a).

1. Set is the category whose objects are sets, and whose arrows are the functions from one set to another.

2. nCob is the category whose objects are closed (n − 1)-dimensional manifolds Σ, and whose arrows M : Σ1 → Σ2 are cobordisms, that is, n-dimensional manifolds having an input boundary Σ1 and an output boundary Σ2.

3. Hilb is the category whose objects are Hilbert spaces and whose arrows are the bounded linear operators from one Hilbert space to another.

The ‘new philosophy’ amounts to the following observation: The last two categories, nCob and Hilb, resemble each other far more than they do the first category, Set! If we loosely regard general relativity or geometry to be represented by nCob, and quantum mechanics to be represented by Hilb, then perhaps many of the difficulties in a theory of quantum gravity, and indeed in quantum mechanics itself, arise due to our silly insistence of thinking of these categories as similar to Set, when in fact the one should be viewed in terms of the other. That is, the notion of points and sets, while mathematically acceptable, might be highly unnatural to the subject at hand!

Automorphisms. Note Quote.

A group automorphism is an isomorphism from a group to itself. If  is a finite multiplicative group, an automorphism of  can be described as a way of rewriting its multiplication table without altering its pattern of repeated elements. For example, the multiplication table of the group of 4th roots of unity  can be written as shown above, which means that the map defined by

is an automorphism of .

Looking at classical geometry and mechanics, Weyl followed Newton and Helmholtz in considering congruence as the basic relation which lay at the heart of the “art of measuring” by the handling of that “sort of bodies we call rigid”. He explained how the local congruence relations established by the comparison of rigid bodies can be generalized and abstracted to congruences of the whole space. In this respect Weyl followed an empiricist approach to classical physical geometry, based on a theoretical extension of the material practice with rigid bodies and their motions. Even the mathematical abstraction to mappings of the whole space carried the mark of their empirical origin and was restricted to the group of proper congruences (orientation preserving isometries of Euclidean space, generated by the translations and rotations) denoted by him as ∆+. This group seems to express “an intrinsic structure of space itself; a structure stamped by space upon all the inhabitants of space”.

But already on the earlier level of physical knowledge, so Weyl argued, the mathematical automorphisms of space were larger than ∆. Even if one sees “with Newton, in congruence the one and only basic concept of geometry from which all others derive”, the group Γ of automorphisms in the mathematical sense turns out to be constituted by the similarities.

The structural condition for an automorphism C ∈ Γ of classical congruence geometry is that any pair (v1,v2) of congruent geometric configurations is transformed into another pair (v1*,v2*) of congruent configurations (vj* = C(vj), j = 1,2). For evaluating this property Weyl introduced the following diagram:

Because of the condition for automorphisms just mentioned the maps C T C-1 and C-1TC belong to ∆+ whenever T does. By this argument he showed that the mathematical automorphism group Γ is the normalizer of the congruences ∆+ in the group of bijective mappings of Euclidean space.

More generally, it also explains the reason for his characterization of generalized similarities in his analysis of the problem of space in the early 1920s. In 1918 he translated the relationship between physical equivalences as congruences to the mathematical automorphisms as the similarities/normalizer of the congruences from classical geometry to special relativity (Minkowski space) and “localized” them (in the sense of physics), i.e., he transferred the structural relationship to the infinitesimal neighbourhoods of the differentiable manifold characterizing spacetime (in more recent language, to the tangent spaces) and developed what later would be called Weylian manifolds, a generalization of Riemannian geometry. In his discussion of the problem of space he generalized the same relationship even further by allowing any (closed) sub-group of the general linear group as a candidate for characterizing generalized congruences at every point.

Moreover, Weyl argued that the enlargement of the physico-geometrical automorphisms of classical geometry (proper congruences) by the mathematical automorphisms (similarities) sheds light on Kant’s riddle of the “incongruous counterparts”. Weyl presented it as the question: Why are “incongruous counterparts” like the left and right hands intrinsically indiscernible, although they cannot be transformed into another by a proper motion? From his point of view the intrinsic indiscernibility could be characterized by the mathematical automorphisms Γ. Of course, the congruences ∆ including the reflections are part of the latter, ∆ ⊂ Γ; this implies indiscernibility between “left and right” as a special case. In this way Kant’s riddle was solved by a Leibnizian type of argument. Weyl very cautiously indicated a philosophical implication of this observation:

And he (Kant) is inclined to think that only transcendental idealism is able to solve this riddle. No doubt, the meaning of congruence and similarity is founded in spatial intuition. Kant seems to aim at some subtler point. But just this point is one which can be completely clarified by general concepts, namely by subsuming it under the general and typical group-theoretic situation explained before . . . .

Weyl stopped here without discussing the relationship between group theoretical methods and the “subtler point” Kant aimed at more explicitly. But we may read this remark as an indication that he considered his reflections on automorphism groups as a contribution to the transcendental analysis of the conceptual constitution of modern science. In his book on Symmetry, he went a tiny step further. Still with the Weylian restraint regarding the discussion of philosophical principles he stated: “As far as I see all a priori statements in physics have their origin in symmetry” (126).

To prepare for the following, Weyl specified the subgroup ∆o ⊂ ∆ with all those transformations that fix one point (∆o = O(3, R), the orthogonal group in 3 dimensions, R the field of real numbers). In passing he remarked:

In the four-dimensional world the Lorentz group takes the place of the orthogonal group. But here I shall restrict myself to the three-dimensional space, only occasionally pointing to the modifications, the inclusion of time into the four-dimensional world brings about.

Keeping this caveat in mind (restriction to three-dimensional space) Weyl characterized the “group of automorphisms of the physical world”, in the sense of classical physics (including quantum mechanics) by the combination (more technically, the semidirect product ̧) of translations and rotations, while the mathematical automorphisms arise from a normal extension:

– physical automorphisms ∆ ≅ R3 X| ∆o with ∆o ≅ O(3), respectively ∆ ≅ R4 X| ∆o for the Lorentz group ∆o ≅ O(1, 3),

– mathematical automorphisms Γ = R+ X ∆
(R+ the positive real numbers with multiplication).

In Weyl’s view the difference between mathematical and physical automorphisms established a fundamental distinction between mathematical geometry and physics.

Congruence, or physical equivalence, is a geometric concept, the meaning of which refers to the laws of physical phenomena; the congruence group ∆ is essentially the group of physical automorphisms. If we interpret geometry as an abstract science dealing with such relations and such relations only as can be logically defined in terms of the one concept of congruence, then the group of geometric automorphisms is the normalizer of ∆ and hence wider than ∆.

He considered this as a striking argument against what he considered to be the Cartesian program of a reductionist geometrization of physics (physics as the science of res extensa):

According to this conception, Descartes’s program of reducing physics to geometry would involve a vicious circle, and the fact that the group of geometric automorphisms is wider than that of physical automorphisms would show that such a reduction is actually impossible.”

In this Weyl alluded to an illusion he himself had shared for a short time as a young scientist. After the creation of his gauge geometry in 1918 and the proposal of a geometrically unified field theory of electromagnetism and gravity he believed, for a short while, to have achieved a complete geometrization of physics.

He gave up this illusion in the middle of the 1920s under the impression of the rising quantum mechanics. In his own contribution to the new quantum mechanics groups and their linear representations played a crucial role. In this respect the mathematical automorphisms of geometry and the physical automorphisms “of Nature”, or more precisely the automorphisms of physical systems, moved even further apart, because now the physical automorphism started to take non-geometrical material degrees of freedom into account (phase symmetry of wave functions and, already earlier, the permutation symmetries of n-particle systems).

But already during the 19th century the physical automorphism group had acquired a far deeper aspect than that of the mobility of rigid bodies:

In physics we have to consider not only points but many types of physical quantities such as velocity, force, electromagnetic field strength, etc. . . .

All these quantities can be represented, relative to a Cartesian frame, by sets of numbers such that any orthogonal transformation T performed on the coordinates keeps the basic physical relations, the physical laws, invariant. Weyl accordingly stated:

All the laws of nature are invariant under the transformations thus induced by the group ∆. Thus physical relativity can be completely described by means of a group of transformations of space-points.

By this argumentation Weyl described a deep shift which ocurred in the late 19th century for the understanding of physics. He described it as an extension of the group of physical automorphisms. The laws of physics (“basic relations” in his more abstract terminology above) could no longer be directly characterized by the motion of rigid bodies because the physics of fields, in particular of electric and magnetic fields, had become central. In this context, the motions of material bodies lost their epistemological primary status and the physical automorphisms acquired a more abstract character, although they were still completely characterizable in geometric terms, by the full group of Euclidean isometries. The indistinguishability of left and right, observed already in clear terms by Kant, acquired the status of a physical symmetry in electromagnetism and in crystallography.

Weyl thus insisted that in classical physics the physical automorphisms could be characterized by the group ∆ of Euclidean isometries, larger than the physical congruences (proper motions) ∆+ but smaller than the mathe- matical automorphisms (similarities) Γ.

This view fitted well to insights which Weyl drew from recent developments in quantum physics. He insisted – differently to what he had thought in 1918 – on the consequence that “length is not relative but absolute” (Hs, p. 15). He argued that physical length measurements were no longer dependent on an arbitrary chosen unit, like in Euclidean geometry. An “absolute standard of length” could be fixed by the quantum mechanical laws of the atomic shell:

The atomic constants of charge and mass of the electron atomic constants and Planck’s quantum of action h, which enter the universal field laws of nature, fix an absolute standard of length, that through the wave lengths of spectral lines is made available for practical measurements.

Space-Time Foliation and Frozen Formalism. Note Quote.

The idea is that one foliates space-time into space and time and considers as fundamental canonical variables the three metric qab and as canonically conjugate momentum a quantity that is closely related to the extrinsic curvature Kab. The time-time and the space-time portions of the space-time metric (known as the lapse and shift vector) appear as Lagrange multipliers in the action, which means that the theory has constraints. In total there are four constraints, that structure themselves into a vector and a scalar. These constraints are the imprint in the canonical theory of the diffeomorphism invariance of the four-dimensional theory. They also contain the dynamics of the theory, since the Hamiltonian identically vanishes. This is not surprising, it is the way in which the canonical formalism tells us that the split into space and time that we perform is a fiduciary one. If one attempts to quantize this theory one starts by choosing a polarization for the wavefunctions (usually functions of the three metric) and one has to implement the constraints as operator equations. These will assure that the wavefunctions embody the symmetries of the theory. The diffeomorphism constraint has a geometrical interpretation, demanding that the wavefunctions be functions of the “three-geometry” and not of the three-metric, that is, that they be invariant under diffeomorphisms of the three manifold. The Hamiltonian constraint does not admit a simple geometric interpretation and should be implemented as an operatorial equation. Unfortunately, it is a complicated non-polynomial function of the basic variables and little progress had been made towards realizing it as a quantum operator ever since De Witt considered the problem in the 60’s. Let us recall that in this context regularization is a highly non-trivial process, since most common regulators used in quantum field theory violate diffeomorphism invariance. Even if we ignore these technical details, the resulting theory appears as very difficult to interpret. The theory has no explicit dynamics, one is in the “frozen formalism”. Wavefunctions are annihilated by the constraints and observable quantities commute with the constraints. Observables are better described, as Kuchar emphasizes, as “perennials”. The expectation is that in physical situations some of the variables of the theory will play the role of “time” and in terms of them one would be able to define a “true” dynamics in a relational way, and a non-vanishing Hamiltonian. In contrast to superstring theory, canonical quantum gravity seeks a non-perturbative quantum theory of only the gravitational field. It aims for consistency between quantum mechanics and gravity, not unification of all the different fields. The main idea is to apply standard quantization procedures to the general theory of relativity. To apply these procedures, it is necessary to cast general relativity into canonical (Hamiltonian) form, and then quantize in the usual way. This was partially successfully done by Dirac. Since it puts relativity into a more familiar form, it makes an otherwise daunting task seem hard but manageable.

Philosophy of Quantum Entanglement and Topology

Many-body entanglement is essential for the existence of topological order in condensed matter systems and understanding many-body entanglement provides a promising approach to understand in general what topological orders exist. It also leads to tensor network descriptions of many-body wave functions potentializing the classification of phases of quantum matter. The generic many-body entanglement is reduced to specifically 2-body systems for choice of entanglement. Consider the equation,

S(A) ≡ −tr(ρA log2A)) —– (1)

where, ρA ≡ trBAB ⟩⟨ΨAB | is the density matrix for part A, and where we assumed that the whole system is in a pure state AB.

Specializing AB⟩ to a ground state in a local Hamiltonian in D dimensions spatially, the central observation being that the entanglement between of a region A of size LD and the (much larger) rest B of the lattice is then often proportional to the size |σ(A)| of the boundary σ(A) of region A,

S(A) ≈ |σ(A)| ≈ LD−1  —– (2)

where, the correction -1 is due to the topological order of the topic code, thus signifying adherence to Boundary Law observed in the ground state of gapped local Hamiltonian in arbitrary dimension D, as well as in some gapless systems in D > 1 dimensions. Instead, in gapless systems in D = 1 dimensions, as well as in certain gapless systems in D > 1 dimensions (namely systems with a Fermi surface of dimension D − 1), ground state entanglement displays a logarithmic correction to the boundary law,

S(A) ≈ |σ(A)| log2 (|σ(A)|) ≈ LD−1 log2(L) —– (3)

At an intuitive level, the boundary law of (2) is understood as resulting from entanglement that involves degrees of freedom located near the boundary between regions A and B. Also intuitively, the logarithmic correction of (3) is argued to have its origin in contributions to entanglement from degrees of freedom that are further away from the boundary between A and B. Given the entanglement between A and B, introducing an entanglement contour sA that assigns a real number sA(i) ≥ 0 to each lattice site i contained in region A such that the sum of sA(i) over all the sites i ∈ A is equal to the entanglement entropy S (A),

S(A) = Σi∈A sA(i) —– (4)

and that aims to quantifying how much the degrees of freedom in site i participate in/contribute to the entanglement between A and B. And as Chen and Vidal put it, the entanglement contour sA(i) is not equivalent to the von Neumann entropy S(i) ≡ −tr ρ(i) log2 ρ(i) of the reduced density matrix ρ(i) at site i. Notice that, indeed, the von Neumann en- tropy of individual sites in region A is not additive in the presence of correlations between the sites, and therefore generically

S(A) ≠ Σi∈A S(i)

whereas the entanglement contour sA(i) is required to fulfil (4). Relatedly, when site i is only entangled with neighboring sites contained within region A, and it is thus uncorrelated with region B, the entanglement contour sA(i) will be required to vanish, whereas the one-site von Neumann entropy S(i) still takes a non-zero value due to the presence of local entanglement within region A.

As an aside, in the traditional approach to quantum mechanics, a physical system is described in a Hilbert space: Observables correspond to self-adjoint operators and statistical operators are associated with the states. In fact, a statistical operator describes a mixture of pure states. Pure states are the really physical states and they are given by rank one statistical operators, or equivalently by rays of the Hilbert space. Von Neumann associated an entropy quantity to a statistical operator and his argument was a gedanken experiment on the ground of phenomenological thermodynamics. Let us consider a gas of N(≫ 1) molecules in a rectangular box K. Suppose that the gas behaves like a quantum system and is described by a statistical operator D, which is a mixture λ|φ1⟩⟨φ1| + (1 − λ)|φ1⟩⟨φ2|, |φi⟩ ≡ φ is a state vector (i = 1, 2). We may take λN molecules in the pure state φ1 and (1−λ)N molecules in the pure state φ2. On the basis of phenomenological thermodynamics, we assume that if φ1 and φ2 are orthogonal, then there is a wall that is completely permeable for the φ1-molecules and isolating for the φ2-molecules. We add an equally large empty rectangular box K′ to the left of the box K and we replace the common wall with two new walls. Wall (a), the one to the left is impenetrable, whereas the one to the right, wall (b), lets through the φ1-molecules but keeps back the φ2-molecules. We add a third wall (c) opposite to (b) which is semipermeable, transparent for the φ2-molecules and impenetrable for the φ1-ones. Then we push slowly (a) and (c) to the left, maintaining their distance. During this process the φ1-molecules are pressed through (b) into K′ and the φ2-molecules diffuse through wall (c) and remain in K. No work is done against the gas pressure, no heat is developed. Replacing the walls (b) and (c) with a rigid absolutely impenetrable wall and removing (a) we restore the boxes K and K′ and succeed in the separation of the φ1-molecules from the φ2-ones without any work being done, without any temperature change and without evolution of heat. The entropy of the original D-gas ( with density N/V ) must be the sum of the entropies of the φ1- and φ2-gases ( with densities λ N/V and (1 − λ)N/V , respectively). If we compress the gases in K and K′ to the volumes λV and (1 − λ)V , respectively, keeping the temperature T constant by means of a heat reservoir, the entropy change amounts to κλN log λ and κ(1 − λ)N log(1 − λ), respectively. Indeed, we have to add heat in the amount of λiNκT logλi (< 0) when the φi-gas is compressed, and dividing by the temperature T we get the change of entropy. Finally, mixing the φ1- and φ2-gases of identical density we obtain a D-gas of N molecules in a volume V at the original temperature. If S0(ψ,N) denotes the entropy of a ψ-gas of N molecules (in a volume V and at the given temperature), we conclude that

S0(φ1,λN)+S0(φ2,(1−λ)N) = S0(D, N) + κλN log λ + κ(1 − λ)N log(1 − λ) —– (5)

must hold, where κ is Boltzmann’s constant. Assuming that S0(ψ,N) is proportional to N and dividing by N we have

λS(φ1) + (1 − λ)S(φ2) = S(D) + κλ log λ + κ(1 − λ) log(1 − λ) —– (6)

where S is certain thermodynamical entropy quantity ( relative to the fixed temperature and molecule density ). We arrived at the mixing property of entropy, but we should not forget about the initial assumption: φ1 and φ2 are supposed to be orthogonal. Instead of a two-component mixture, von Neumann operated by an infinite mixture, which does not make a big difference, and he concluded that

S (Σiλi|φi⟩⟨φi|) = ΣiλiS(|φi⟩⟨φi|) − κ Σiλi log λi —– (7)

Von Neumann’s argument does not require that the statistical operator D is a mixture of pure states. What we really needed is the property D = λD1 + (1 − λ)D2 in such a way that the possible mixed states D1 and D2 are disjoint. D1 and D2 are disjoint in the thermodynamical sense, when there is a wall which is completely permeable for the molecules of a D1gas and isolating for the molecules of a D2-gas. In other words, if the mixed states D1 and D2 are disjoint, then this should be demonstrated by a certain filter. Mathematically, the disjointness of D1 and D2 is expressed in the orthogonality of the eigenvectors corresponding to nonzero eigenvalues of the two density matrices. The essential point is in the remark that (6) must hold also in a more general situation when possibly the states do not correspond to density matrices, but orthogonality of the states makes sense:

λS(D1) + (1 − λ)S(D2) = S(D) + κλ log λ + κ(1 − λ) log(1 − λ) —– (8)

(7) reduces the determination of the (thermodynamical) entropy of a mixed state to that of pure states. The so-called Schatten decomposition Σi λi|φi⟩⟨φi| of a statistical operator is not unique even if ⟨φi , φj ⟩ = 0 is assumed for i ≠ j . When λi is an eigenvalue with multiplicity, then the corresponding eigenvectors can be chosen in many ways. If we expect the entropy S(D) to be independent of the Schatten decomposition, then we are led to the conclusion that S(|φ⟩⟨φ|) must be independent of the state vector |φ⟩. This argument assumes that there are no superselection sectors, that is, any vector of the Hilbert space can be a state vector. On the other hand, von Neumann wanted to avoid degeneracy of the spectrum of a statistical operator. Von Neumann’s proof of the property that S(|φ⟩⟨φ|) is independent of the state vector |φ⟩ was different. He did not want to refer to a unitary time development sending one state vector to another, because that argument requires great freedom in choosing the energy operator H. Namely, for any |φ1⟩ and |φ2⟩ we would need an energy operator H such that

eitH|φ1⟩ = |φ2⟩

This process would be reversible. Anyways, that was quite a digression.

Entanglement between A and B is naturally described by the coefficients {pα} appearing in the Schmidt decomposition of the state |ΨAB⟩,

AB⟩ = Σα √pαAα ⟩ ⊗ |ΨBα ⟩ —– (9)

These coefficients {pα} correspond to the eigenvalues of the reduced density matrix ρA, whose spectral decomposition reads

ρA = ΣαpAα⟩⟨ΨAα—– (10)

defining a probability distribution, pα ≥ 0, Σα pα = 1, in terms of which the von Neumann entropy S(A) is

S(A) = − Σαpα log2(pα—– (11)

On the other hand, the Hilbert space VA of region A factorizes as the tensor product

VA = ⊗ i∈A V(i) —– (12)

where V(i) describes the local Hilbert space of site i. The reduced density matrix ρA in (10) and the factorization of (12) define two inequivalent structures within the vector space VA of region A. The entanglement contours A is a function from the set of sites i∈A to the real numbers,

sA : A → ℜ —– (13)

that attempts to relate these two structures, by distributing the von-Neumann entropy S(A) of (11) among the sites i ∈ A. According to Chen and Vidal, there are five conditions/requirements on entanglement contours that need satiation.

a. Positivity: sA(i) ≥ 0

b. Normalization: Σi∈AsA(i) = S(A)

These constraints amount to defining a probability distribution pi ≡ sA(i)/S(A) over the sites i ∈ A, with pi ≥ 0 and i Σipi = 1, such that sA(i) = piS(A), however, do not requiring sA to inform us about the spatial structure of entanglement in A, but only relating to the density matrix ρA through its total von Neumann entropy S(A).

c. Symmetry: if T is a symmetry of ρA, that is AT = ρA, and T exchanges site i with site j, then sA(i) = sA(j).

This condition ensures that the entanglement contour is the same on two sites i and j of region A that, as far as entanglement is concerned, play an equivalent role in region A. It uses the (possible) presence of a spatial symmetry, such as invariance under space reflection, or under discrete translations/rotations, to define an equivalence relation in the set of sites of region A, and requires that the entanglement contour be constant within each resulting equivalence class. Notice, however, that this condition does not tell us whether the entanglement contour should be large or small on a given site (or equivalence class of site). In particular, the three conditions above are satisfied by a canonical choice sA(i) = S (A)/|A|, that is a flat entanglement contour over the |A| sites contained in region A, which once more does not tell us anything about the spatial structure of the von Neumann entropy in ρA.

The remaining conditions refer to subregions within region A, instead of referring to single sites. It is therefore convenient to (trivially) extend the definition of entanglement contour to a set X of sites in region A, X ⊆ A, with vector space

VX = ⊗i∈X V(i) —– (14)

as the sum of the contour over the sites in X,

sA(X) ≡  Σi∈XsA(i) —– (15)

It follows from this extension that for any two disjoint subsets X1, X2 ⊆ A, with X1 ∩ X2 = ∅, the contour is additive,

sA(X1 ∪ X2) = sA(X1) + sA(X2—– (16)

In particular, condition 2 can be now recast as sA(A) =S(A). Similarly, if X, X ⊆ A, are such that all the sites of X1 are also contained in X2, X1X2 ,then the contour must be larger on X2 than on X1 (monotonicity of sA(X)),

sA(X1) ≤ sA(X2) if X1 ⊆ X2 —– (17)

d. Invariance under local unitary transformations: if the state |Ψ′AB is obtained from the state AB by means of a unitary transformation UX that acts on a subset X ⊆ A of sites of region A, that is |Ψ′AB⟩ ≡ UXAB, then the entanglement contour sA(X) must be the same for state AB and for state |Ψ′AB.

That is, the contribution of region X to the entanglement between A and B is not affected by a redefinition of the sites or change of basis within region X. Notice that it follows that  Ucan also not change sA(X’), where X’ ≡ A − X is the complement of X in A.

To motivate our last condition, let us consider a state AB that factorizes as the product

AB⟩ = |ΨXXB⟩ ⊗ |ΨX’X’B—– (18)

where X ⊆ A and XB ⊆ B are subsets of sites in regions A and B, respectively, and X’ ⊆ A and X’B ⊆ B are their complements within A and B, so that

VA = VX ⊗ VX’, —– (19)

VB = VXB ⊗ VX’B —– (20)

in this case the reduced density matrix ρA factorizes as ρA = ρX ⊗ ρX’ and the entanglement entropy is additive,

S(A) = S(X) + S(X’) —– (21)

Since the entanglement entropy S(X) of subregion X is well-defined, let the entanglement profile over X be equal to it,

sA(X) = S(X) —– (22)

The last condition refers to a more general situation where, instead of obeying (18), the state AB factorizes as the product

AB⟩ = |ΨΩAΩB⟩ ⊗ |ΨΩ’AΩ’B, —– (23)

with respect to some decomposition of VA and VB as

tensor products of factor spaces,

VA = VΩA ⊗ VΩ’A, —– (24)

VB = VΩB ⊗ VΩ’B —– (25)

Let S(ΩA) denote the entanglement entropy supported on the first factor space VΩA of  VA, that is

S(ΩA) = −tr(ρΩA log2ΩA)) —– (26)

ρΩA ≡ trΩB |Ψ ΩA ΩB⟩⟨Ψ ΩA ΩB| —– (27)

and let X ⊆ A be a subset of sites whose vector space VX is completely contained in VΩA , meaning that VΩA can be further decomposed as

VΩA  ≈ VX VX’ —– (28)

e. Upper bound: if a subregion X ⊆ A is contained in a factor space ΩA (24 and 28) then the entanglement contour of subregion X cannot be larger than the entanglement entropy S(ΩA) (26)

sA(X) S(ΩA) —– (29)

This condition says that whenever we can ascribe a concrete value S(ΩA) of the entanglement entropy to a factor space ΩA within region A (that is, whenever the state AB factorizes as in (24) then the entanglement contour has to be consistent with this fact, meaning that the contour S(X) in any subregion X contained in the factor space ΩA is upper bounded by S(ΩA).

Let us consider a particular case of condition e. When a region X ∈ A is not at all correlated with B, that is ρXBX ⊗ ρB,then it can be seen that X is contained in some factor space ΩA such that the state |Ψ ΩA ΩB itself further factorizes as |Ψ ΩA⟩ ⊗ |ΨΩB, so that (23) becomes

AB⟩ = |Ψ ΩA⟩ ⊗ |ΨΩB ⊗ |ΨΩ’AΩ’B ⟩, —– (30)

and S(ΩA) = 0. Condition e then requires that sA(X) = 0, that is

ρXBX ⊗ ρB sA(X) = 0, —– (31)

reflecting the fact that a region X ⊆ A that is not correlated with B does not contribute at all to the entanglement between A and B. Finally, the upper bound in e can be alternatively announced as a lower bound. Let Y ⊆ A be a subset of sites whose vector space VY completely contains VΩA in (24), meaning that VY can be further decomposed as

VY VΩA ⊗ VΩ’A —– (32)

e’. Lower bound: The entanglement contour of subregion Y is at least equal to the entanglement entropy S(ΩA) in (26),

sA(Y) ≥ S(ΩA) —– (33)

Conditions a-e (e’) are not expected to completely determine the entanglement contour. In other words, there probably are inequivalent functions sA : A → ℜ that conform to all the conditions above. So, where do we get philosophical from here? It is through the entanglement contour through selected states that a time evolution ensuing a global or a local quantum quench characterizing entanglement between regions rather than within regions, revealing a a detailed real-space structure of the entanglement of a region A and its dynamics, well beyond what is accessible from the entanglement entropy alone. But, that isn’t all. Questions of how to quantify entanglement and non-locality, and the need to clarify the relationship between them are important not only conceptually, but also practically, insofar as entanglement and non-locality seem to be different resources for the performance of quantum information processing tasks. Whether in a given quantum information protocol (cryptography, teleportation, and algorithm . . .) it is better to look for the largest amount of entanglement or the largest amount of non-locality becomes decisive. The ever-evolving field of quantum information theory is devoted to using the principles and laws of quantum mechanics to aid in the acquisition, transmission, and processing of information. In particular, it seeks to harness the peculiarly quantum phenomena of entanglement, superposition, and non-locality to perform all sorts of novel tasks, such as enabling computations that operate exponentially faster or more efficiently than their classical counterparts (via quantum computers) and providing unconditionally secure cryptographic systems for the transfer of secret messages over public channels (via quantum key distribution). By contrast, classical information theory is concerned with the storage and transfer of information in classical systems. It uses the “bit” as the fundamental unit of information, where the system capable of representing a bit can take on one of two values (typically 0 or 1). Classical information theory is based largely on the concept of information formalized by Claude Shannon in the late 1940s. Quantum information theory, which was later developed in analogy with classical information theory, is concerned with the storage and processing of information in quantum systems, such as the photon, electron, quantum dot, or atom. Instead of using the bit, however, it defines the fundamental unit of quantum information as the “qubit.” What makes the qubit different from a classical bit is that the smallest system capable of storing a qubit, the two-level quantum system, not only can take on the two distinct values |0 and |1 , but can also be in a state of superposition of these two states: |ψ = α0 |0 + α1 |1.

Quantum information theory has opened up a whole new range of philosophical and foundational questions in quantum cryptography or quantum key distribution, which involves using the principles of quantum mechanics to ensure secure communication. Some quantum cryptographic protocols make use of entanglement to establish correlations between systems that would be lost upon eavesdropping. Moreover, a quantum principle known as the no-cloning theorem prohibits making identical copies of an unknown quantum state. In the context of a C∗-algebraic formulation,  quantum theory can be characterized in terms of three information-theoretic constraints: (1) no superluminal signaling via measurement, (2) no cloning (for pure states) or no broadcasting (mixed states), and (3) no unconditionally secure bit commitment.

Entanglement does not refute the principle of locality. A sketch of the sort of experiment commonly said to refute locality runs as follows. Suppose that you have two electrons with entangled spin. For each electron you can measure the spin along the X, Y or Z direction. If you measure X on both electrons, then you get opposite values, likewise for measuring Y or Z on both electrons. If you measure X on one electron and Y or Z on the other, then you have a 50% probability of a match. And if you measure Y on one and Z on the other, the probability of a match is 50%. The crucial issue is that whether you find a correlation when you do the comparison depends on whether you measure the same quantity on each electron. Bell’s theorem just explains that the extent of this correlation is greater than a local theory would allow if the measured quantities were represented by stochastic variables (i.e. – numbers picked out of a hat). This fact is often misrepresented as implying that quantum mechanics is non-local. But in quantum mechanics, systems are not characterised by stochastic variables, but, rather, by Hermitian operators. There is an entirely local explanation of how the correlations arise in terms of properties of systems represented by such operators. But, another answer to such violations of the principle of locality could also be “Yes, unless you get really obsessive about it.” It has been formally proven that one can have determinacy in a model of quantum dynamics, or one can have locality, but cannot have both. If one gives up the determinacy of the theory in various ways, one can imagine all kinds of ‘planned flukes’ like the notion that the experiments that demonstrate entanglement leak information and pre-determine the environment to make the coordinated behavior seem real. Since this kind of information shaping through distributed uncertainty remains a possibility, folks can cling to locality until someone actually manages something like what those authors are attempting, or we find it impossible. If one gives up locality instead, entanglement does not present a problem, the theory of relativity does. Because the notion of a frame of reference is local. Experiments on quantum tunneling that violate the constraints of the speed of light have been explained with the idea that probabilistic partial information can ‘lead’ real information faster than light by pushing at the vacuum underneath via the ‘Casimir Effect’. If both of these make sense, then the information carried by the entanglement when it is broken would be limited as the particles get farther apart — entanglements would have to spontaneously break down over time or distance of separation so that the probabilities line up. This bodes ill for our ability to find entangled particles from the Big Bang, which seems to be the only prospect in progress to debunk the excessively locality-focussed.

But, much of the work remains undone and this is to be continued…..