# Gauge Fixity Towards Hyperbolicity: The Case For Equivalences. Part 2. The Lagrangian has in fact to depend on reference backgrounds in a quite peculiar way, so that a reference background cannot interact with any other physical field, otherwise its effect would be observable in a laboratory….

Let then Γ’ be any (torsionless) reference connection. Introducing the following relative quantities, which are both tensors:

qμαβ = Γμαβ – Γ’μαβ

wμαβ = uμαβ – u’μαβ —– (1)

For any linear torsionless connection Γ’, the Hilbert-Einstein Lagrangian

LH: J2Lor(m) → ∧om(M)

LH: LH(gαβ, Rαβ)ds = 1/2κ (R – 2∧)√g ds

can be covariantly recast as:

LH = dα(Pβμuαβμ)ds + 1/2κ[gβμρβσΓσρμ – ΓαασΓσβμ) – 2∧]√g ds

= dα(Pβμwαβμ)ds + 1/2κ[gβμ(R’βμ + qρβσqσρμ – qαασqσβμ)  – 2∧]√g ds —– (2)

The first expression for LH shows that Γ’ (or g’, if Γ’ are assumed a priori to be Christoffel symbols of the reference metric g’) has no dynamics, i.e. field equations for the reference connection are identically satisfied (since any dependence on it is hidden under a divergence). The second expression shows instead that the same Einstein equations for g can be obtained as the Euler-Lagrange equation for the Lagrangian:

L1 = 1/2κ[gβμ(R’βμ + qρβσqσρμ – qαασqσβμ)  – 2∧]√g ds —– (3)

which is first order in the dynamical field g and it is covariant since q is a tensor. The two Lagrangians Land L1, are thence said to be equivalent, since they provide the same field equations.

In order to define the natural theory, we will have to declare our attitude towards the reference field Γ’. One possibility is to mimic the procedure used in Yang-Mills theories, i.e. restrict to variations which keep the reference background fixed. Alternatively we can consider Γ’ (or g’) as a dynamical field exactly as g is, even though the reference is not endowed with a physical meaning. In other words, we consider arbitrary variations and arbitrary transformations even if we declare that g is “observable” and genuinely related to the gravitational field, while Γ’ is not observable and it just sets the reference level of conserved quantities. A further important role played by Γ’ is that it allows covariance of the first order Lagrangian L1, . No first order Lagrangian for Einstein equations exists, in fact, if one does not allow the existence of a reference background field (a connection or something else, e.g. a metric or a tetrad field). To obtain a good and physically sound theory out of the Lagrangian L1, we still have to improve its dependence on the reference background Γ’. For brevity’s sake, let us assume that Γ’ is the Levi-Civita connection of a metric g’ which thence becomes the reference background. Let us also assume (even if this is not at all necessary) that the reference background g’ is Lorentzian. We shall introduce a dynamics for the reference background g’, (thus transforming its Levi-Civita connection into a truly dynamical connection), by considering a new Lagrangian:

L1B = 1/2κ[√g(R – 2∧) – dα(√g gμνwαμν) – √g'(R’ – 2∧)]ds

= 1/2κ[(R’ – 2∧)(√g – √g’) + √g gβμ(qρβσqσρμ – qαασqσβμ)]ds —– (4)

which is obtained from L1 by subtracting the kinetic term (R’ – 2∧) √g’. The field g’ is no longer undetermined by field equations, but it has to be a solution of the variational equations for L1B w. r. t. g, which coincide with Einstein field equations. Why should a reference field, which we pretend not to be observable, obey some field equation? Field equations are here functional to the role that g’ plays in our framework. If g’ has to fix the zero value of conserved quantities of g which are relative to the reference configuration g’ it is thence reasonable to require that g’ is a solution of Einstein equations as well. Under this assumption, in fact, both g and g’ represent a physical situation and relative conserved quantities represent, for example, the energy “spent to go” from the configuration g’ to the configuration g. To be strictly precise, further hypotheses should be made to make the whole matter physically meaningful in concrete situations. In a suitable sense we have to ensure that g’ and g belong to the same equivalence class under some (yet undetermined equivalence relation), e.g. that g’ can be homotopically deformed onto g or that they satisfy some common set of boundary (or asymptotic) conditions.

Considering the Lagrangian L1B as a function of the two dynamical fields g and g’, first order in g and second order in g’. The field g is endowed with a physical meaning ultimately related to the gravitational field, while g’ is not observable and it provides at once covariance and the zero level of conserved quantities. Moreover, deformations will be ordinary (unrestricted) deformations both on g’ and g, and symmetries will drag both g’ and g. Of course, a natural framework has to be absolute to have a sense; any further trick or limitation does eventually destroy the naturality. The Lagrangian L1B is thence a Lagrangian

L1B : J2Lor(M) xM J1Lor(M) → Am(M)

# Game Theory and Finite Strategies: Nash Equilibrium Takes Quantum Computations to Optimality. Finite games of strategy, within the framework of noncooperative quantum game theory, can be approached from finite chain categories, where, by finite chain category, it is understood a category C(n;N) that is generated by n objects and N morphic chains, called primitive chains, linking the objects in a specific order, such that there is a single labelling. C(n;N) is, thus, generated by N primitive chains of the form:

x0 →f1 x1 →f2 x1 → … xn-1 →fn xn —– (1)

A finite chain category is interpreted as a finite game category as follows: to each morphism in a chain xi-1 →fi xi, there corresponds a strategy played by a player that occupies the position i, in this way, a chain corresponds to a sequence of strategic choices available to the players. A quantum formal theory, for a finite game category C(n;N), is defined as a formal structure such that each morphic fundament fi of the morphic relation xi-1 →fi xis a tuple of the form:

fi := (Hi, Pi, Pˆfi) —– (2)

where Hi is the i-th player’s Hilbert space, Pi is a complete set of projectors onto a basis that spans the Hilbert space, and Pˆfi ∈ Pi. This structure is interpreted as follows: from the strategic Hilbert space Hi, given the pure strategies’ projectors Pi, the player chooses to play Pˆfi .

From the morphic fundament (2), an assumption has to be made on composition in the finite category, we assume the following tensor product composition operation:

fj ◦ fi = fji —– (3)

fji = (Hji = Hj ⊗ Hi, Pji = Pj ⊗ Pi, Pˆfji = Pˆfj ⊗ Pˆfi) —– (4)

From here, a morphism for a game choice path could be introduced as:

x0 →fn…21 xn —– (5)

fn…21 = (HG = ⊗i=n1 Hi, PG = ⊗i=n1 Pi, Pˆ fn…21 = ⊗i=n1fi) —– (6)

in this way, the choices along the chain of players are completely encoded in the tensor product projectors Pˆfn…21. There are N = ∏i=1n dim(Hi) such morphisms, a number that coincides with the number of primitive chains in the category C(n;N).

Each projector can be addressed as a strategic marker of a game path, and leads to the matrix form of an Arrow-Debreu security, therefore, we call it game Arrow-Debreu projector. While, in traditional financial economics, the Arrow-Debreu securities pay one unit of numeraire per state of nature, in the present game setting, they pay one unit of payoff per game path at the beginning of the game, however this analogy may be taken it must be addressed with some care, since these are not securities, rather, they represent, projectively, strategic choice chains in the game, so that the price of a projector Pˆfn…21 (the Arrow-Debreu price) is the price of a strategic choice and, therefore, the result of the strategic evaluation of the game by the different players.

Now, let |Ψ⟩ be a ket vector in the game’s Hilbert space HG, such that:

|Ψ⟩ = ∑fn…21 ψ(fn…21)|(fn…21⟩ —– (7)

where ψ(fn…21) is the Arrow-Debreu price amplitude, with the condition:

fn…21 |ψ(fn…21)|2 = D —– (8)

for D > 0, then, the |ψ(fn…21)|corresponds to the Arrow-Debreu prices for the game path fn…21 and D is the discount factor in riskless borrowing, defining an economic scale for temporal connections between one unit of payoff now and one unit of payoff at the end of the game, such that one unit of payoff now can be capitalized to the end of the game (when the decision takes place) through a multiplication by 1/D, while one unit of payoff at the end of the game can be discounted to the beginning of the game through multiplication by D.

In this case, the unit operator, 1ˆ = ∑fn…21 Pˆfn…21 has a similar profile as that of a bond in standard financial economics, with ⟨Ψ|1ˆ|Ψ⟩ = D, on the other hand, the general payoff system, for each player, can be addressed from an operator expansion:

πiˆ = ∑fn…21 πi (fn…21) Pˆfn…21 —– (9)

where each weight πi(fn…21) corresponds to quantities associated with each Arrow-Debreu projector that can be interpreted as similar to the quantities of each Arrow-Debreu security for a general asset. Multiplying each weight by the corresponding Arrow-Debreu price, one obtains the payoff value for each alternative such that the total payoff for the player at the end of the game is given by:

⟨Ψ|1ˆ|Ψ⟩ = ∑fn…21 πi(fn…21) |ψ(fn…21)|2/D —– (10)

We can discount the total payoff to the beginning of the game using the discount factor D, leading to the present value payoff for the player:

PVi = D ⟨Ψ|πiˆ|Ψ⟩ = D ∑fn…21 π (fn…21) |ψ(fn…21)|2/D —– (11)

, where π (fn…21) represents quantities, while the ratio |ψ(fn…21)|2/D represents the future value at the decision moment of the quantum Arrow- Debreu prices (capitalized quantum Arrow-Debreu prices). Introducing the ket

|Q⟩ ∈ HG, such that:

|Q⟩ = 1/√D |Ψ⟩ —– (12)

then, |Q⟩ is a normalized ket for which the price amplitudes are expressed in terms of their future value. Replacing in (11), we have:

PVi = D ⟨Q|πˆi|Q⟩ —– (13)

In the quantum game setting, the capitalized Arrow-Debreu price amplitudes ⟨fn…21|Q⟩ become quantum strategic configurations, resulting from the strategic cognition of the players with respect to the game. Given |Q⟩, each player’s strategic valuation of each pure strategy can be obtained by introducing the projector chains:

Cˆfi = ∑fn…i+1fi-1…1 Pˆfn…i+1 ⊗ Pˆfi ⊗ Pˆfi-1…1 —– (14)

with ∑fi Cˆfi = 1ˆ. For each alternative choice of the player i, the chain sums over all of the other choice paths for the rest of the players, such chains are called coarse-grained chains in the decoherent histories approach to quantum mechanics. Following this approach, one may introduce the pricing functional from the expression for the decoherence functional:

D (fi, gi : |Q⟩) = ⟨Q| Cˆfi Cgi|Q⟩  —– (15)

we, then have, for each player

D (fi, gi : |Q⟩) = 0, ∀ fi ≠ gi —– (16)

this is the usual quantum mechanics’ condition for an aditivity of measure (also known as decoherence condition), which means that the capitalized prices for two different alternative choices of player i are additive. Then, we can work with the pricing functional D(fi, fi :|Q⟩) as giving, for each player an Arrow-Debreu capitalized price associated with the pure strategy, expressed by fi. Given that (16) is satisfied, each player’s quantum Arrow-Debreu pricing matrix, defined analogously to the decoherence matrix from the decoherent histories approach, is a diagonal matrix and can be expanded as a linear combination of the projectors for each player’s pure strategies as follows:

Di (|Q⟩) = ∑fi D(fi, f: |Q⟩) Pˆfi —– (17)

which has the mathematical expression of a mixed strategy. Thus, each player chooses from all of the possible quantum computations, the one that maximizes the present value payoff function with all the other strategies held fixed, which is in agreement with Nash.

# Vector Fields Tangent to the Surfaces of Foliation. Note Quote. Although we are interested in gauge field theories, we will use mainly the language of mechanics that is, of a finite number of degrees of freedom, which is sufficient for our purposes. A quick switch to the field theory language can be achieved by using DeWitt’s condensed notation. Consider, as our starting point a time-independent first- order Lagrangian L(q, q ̇) defined in configuration-velocity space TQ, that is, the tangent bundle of some configuration manifold Q that we assume to be of dimension n. Gauge theories rely on singular as opposed to regular Lagrangians, that is, Lagrangians whose Hessian matrix with respect to the velocities (where q stands, in a free index notation, for local coordinates in Q),

Wij ≡ ∂2L/∂q.i∂q.j —– (1)

is not invertible. Two main consequences are drawn from this non-invertibility. First notice that the Euler-Lagrange equations of motion [L]i = 0, with

[L]i : = αi − Wijq ̈j

and

αi := ∂2L/∂q.i∂q.j q.j

cannot be written in a normal form, that is, isolating on one side the accelerations q ̈ = f (q, q ̇). This makes the usual theorems about the existence and uniqueness of solutions of ordinary differential equations inapplicable. Consequently, there may be points in the tangent bundle where there are no solutions passing through the point, and others where there is more than one solution.

The second consequence of the Hessian matrix being singular concerns the construction of the canonical formalism. The Legendre map from the tangent bundle TQ to the cotangent bundle —or phase space— T ∗Q (we use the notation pˆ(q, q ̇) := ∂L/∂q ̇),

FL : TQ → T ∗ Q —– (2)

(q, q ̇) → (q, p=pˆ) —– (3)

is no longer invertible because ∂pˆ/∂q ̇ = ∂L/∂q ̇∂q ̇ is the Hessian matrix. There appears then an issue about the projectability of structures from the tangent bundle to phase space: there will be functions defined on TQ that cannot be translated (projected) to functions on phase space. This feature of the formalisms propagates in a corresponding way to the tensor structures, forms, vector fields, etc.

In order to better identify the problem and to obtain the conditions of projectability, we must be more specific. We will make a single assumption, which is that the rank of the Hessian matrix is constant everywhere. If this condition is not satisfied throughout the whole tangent bundle, we will restrict our considerations to a region of it, with the same dimensionality, where this condition holds. So we are assuming that the rank of the Legendre map FL is constant throughout T Q and equal to, say, 2n − k. The image of FL will be locally defined by the vanishing of k independent functions, φμ(q, p), μ = 1, 2, .., k. These functions are the primary constraints, and their pullback FL ∗ φμ to the tangent bundle is identically zero:

(FL ∗ φμ)(q, q ̇) := φμ(q, pˆ) = 0, ∀ q, q ̇—– (4)

The primary constraints form a generating set of the ideal of functions that vanish on the image of the Legendre map. With their help it is easy to obtain a basis of null vectors for the Hessian matrix. Indeed, applying ∂/∂q. to (4) we get

Wij = (∂φμ/∂pj)|p=pˆ = 0, ∀ q, q ̇ —– (5)

With this result in hand, let us consider some geometrical aspects of the Legendre map. We already know that its image in T∗Q is given by the primary constraints’ surface. A foliation in TQ is also defined, with each element given as the inverse image of a point in the primary constraints’ surface in T∗Q. One can easily prove that the vector fields tangent to the surfaces of the foliation are generated by

Γμ= (∂φμ/∂pj)|p=pˆ = ∂/∂q.j —– (6)

The proof goes as follows. Consider two neighboring points in TQ belonging to the same sheet, (q, q ̇) and (q, q ̇ + δq ̇) (the configuration coordinates q must be the same because they are preserved by the Legendre map). Then, using the definition of the Legendre map, we must have pˆ(q, q ̇) = pˆ(q, q ̇ + δq ̇), which implies, expanding to first order,

∂pˆ/ ∂q ̇ δ q ̇ = 0

which identifies δq ̇ as a null vector of the Hessian matrix (here expressed as ∂pˆ/∂q ̇). Since we already know a basis for such null vectors, (∂φμ /∂pj)|p=pˆ, μ = 1, 2, …, k, it follows that the vector fields Γμ form a basis for the vector fields tangent to the foliation.

The knowledge of these vector fields is instrumental for addressing the issue of the projectability of structures. Consider a real-valued function fL: TQ → R. It will — locally— define a function fH: T∗Q −→ R iff it is constant on the sheets of the foliation, that is, when

ΓμfL = 0, μ = 1,2,…,k. (7)

Equation (7) is the projectability condition we were looking for. We express it in the following way:

ΓμfL = 0, μ = 1,2,…,k ⇔ there exists fH such that FL ∗ fH = fL

# Theories of Fields: Gravitational Field as “the More Equal Among Equals” Descartes, in Le Monde, gave a fully relational definition of localization (space) and motion. According to Descartes, there is no “empty space”. There are only objects, and it makes sense to say that an object A is contiguous to an object B. The “location” of an object A is the set of the objects to which A is contiguous. “Motion” is change in location. That is, when we say that A moves we mean that A goes from the contiguity of an object B to the contiguity of an object C3. A consequence of this relationalism is that there is no meaning in saying “A moves”, except if we specify with respect to which other objects (B, C,. . . ) it is moving. Thus, there is no “absolute” motion. This is the same definition of space, location, and motion, that we find in Aristotle. Aristotle insists on this point, using the example of the river that moves with respect to the ground, in which there is a boat that moves with respect to the water, on which there is a man that walks with respect to the boat . . . . Aristotle’s relationalism is tempered by the fact that there is, after all, a preferred set of objects that we can use as universal reference: the Earth at the center of the universe, the celestial spheres, the fixed stars. Thus, we can say, if we desire so, that something is moving “in absolute terms”, if it moves with respect to the Earth. Of course, there are two preferred frames in ancient cosmology: the one of the Earth and the one of the fixed stars; the two rotates with respect to each other. It is interesting to notice that the thinkers of the middle ages did not miss this point, and discussed whether we can say that the stars rotate around the Earth, rather than being the Earth that rotates under the fixed stars. Buridan concluded that, on ground of reason, in no way one view is more defensible than the other. For Descartes, who writes, of course, after the great Copernican divide, the Earth is not anymore the center of the Universe and cannot offer a naturally preferred definition of stillness. According to malignants, Descartes, fearing the Church and scared by what happened to Galileo’s stubborn defense of the idea that “the Earth moves”, resorted to relationalism, in Le Monde, precisely to be able to hold Copernicanism without having to commit himself to the absolute motion of the Earth!

Relationalism, namely the idea that motion can be defined only in relation to other objects, should not be confused with Galilean relativity. Galilean relativity is the statement that “rectilinear uniform motion” is a priori indistinguishable from stasis. Namely that velocity (but just velocity!), is relative to other bodies. Relationalism holds that any motion (however zigzagging) is a priori indistinguishable from stasis. The very formulation of Galilean relativity requires a nonrelational definition of motion (“rectilinear and uniform” with respect to what?).

Newton took a fully different course. He devotes much energy to criticise Descartes’ relationalism, and to introduce a different view. According to him, space exists. It exists even if there are no bodies in it. Location of an object is the part of space that the object occupies. Motion is change of location. Thus, we can say whether an object moves or not, irrespectively from surrounding objects. Newton argues that the notion of absolute motion is necessary for constructing mechanics. His famous discussion of the experiment of the rotating bucket in the Principia is one of the arguments to prove that motion is absolute.

This point has often raised confusion because one of the corollaries of Newtonian mechanics is that there is no detectable preferred referential frame. Therefore the notion of absolute velocity is, actually, meaningless, in Newtonian mechanics. The important point, however, is that in Newtonian mechanics velocity is relative, but any other feature of motion is not relative: it is absolute. In particular, acceleration is absolute. It is acceleration that Newton needs to construct his mechanics; it is acceleration that the bucket experiment is supposed to prove to be absolute, against Descartes. In a sense, Newton overdid a bit, introducing the notion of absolute position and velocity (perhaps even just for explanatory purposes?). Many people have later criticised Newton for his unnecessary use of absolute position. But this is irrelevant for the present discussion. The important point here is that Newtonian mechanics requires absolute acceleration, against Aristotle and against Descartes. Precisely the same does special relativistic mechanics.

Similarly, Newton introduced absolute time. Newtonian space and time or, in modern terms, spacetime, are like a stage over which the action of physics takes place, the various dynamical entities being the actors. The key feature of this stage, Newtonian spacetime, is its metrical structure. Curves have length, surfaces have area, regions of spacetime have volume. Spacetime points are at fixed distance the one from the other. Revealing, or measuring, this distance, is very simple. It is sufficient to take a rod and put it between two points. Any two points which are one rod apart are at the same distance. Using modern terminology, physical space is a linear three-dimensional (3d) space, with a preferred metric. On this space there exist preferred coordinates xi, i = 1,2,3, in terms of which the metric is just δij. Time is described by a single variable t. The metric δij determines lengths, areas and volumes and defines what we mean by straight lines in space. If a particle deviates with respect to this straight line, it is, according to Newton, accelerating. It is not accelerating with respect to this or that dynamical object: it is accelerating in absolute terms.

Special relativity changes this picture only marginally, loosing up the strict distinction between the “space” and the “time” components of spacetime. In Newtonian spacetime, space is given by fixed 3d planes. In special relativistic spacetime, which 3d plane you call space depends on your state of motion. Spacetime is now a 4d manifold M with a flat Lorentzian metric ημν. Again, there are preferred coordinates xμ, μ = 0, 1, 2, 3, in terms of which ημν = diag[1, −1, −1, −1]. This tensor, ημν , enters all physical equations, representing the determinant influence of the stage and of its metrical properties on the motion of anything. Absolute acceleration is deviation of the world line of a particle from the straight lines defined by ημν. The only essential novelty with special relativity is that the “dynamical objects”, or “bodies” moving over spacetime now include the fields as well. Example: a violent burst of electromagnetic waves coming from a distant supernova has traveled across space and has reached our instruments. For the rest, the Newtonian construct of a fixed background stage over which physics happen is not altered by special relativity.

The profound change comes with general relativity (GTR). The central discovery of GR, can be enunciated in three points. One of these is conceptually simple, the other two are tremendous. First, the gravitational force is mediated by a field, very much like the electromagnetic field: the gravitational field. Second, Newton’s spacetime, the background stage that Newton introduced introduced, against most of the earlier European tradition, and the gravitational field, are the same thing. Third, the dynamics of the gravitational field, of the other fields such as the electromagnetic field, and any other dynamical object, is fully relational, in the Aristotelian-Cartesian sense. Let me illustrate these three points.

First, the gravitational field is represented by a field on spacetime, gμν(x), just like the electromagnetic field Aμ(x). They are both very concrete entities: a strong electromagnetic wave can hit you and knock you down; and so can a strong gravitational wave. The gravitational field has independent degrees of freedom, and is governed by dynamical equations, the Einstein equations.

Second, the spacetime metric ημν disappears from all equations of physics (recall it was ubiquitous). At its place – we are instructed by GTR – we must insert the gravitational field gμν(x). This is a spectacular step: Newton’s background spacetime was nothing but the gravitational field! The stage is promoted to be one of the actors. Thus, in all physical equations one now sees the direct influence of the gravitational field. How can the gravitational field determine the metrical properties of things, which are revealed, say, by rods and clocks? Simply, the inter-atomic separation of the rods’ atoms, and the frequency of the clock’s pendulum are determined by explicit couplings of the rod’s and clock’s variables with the gravitational field gμν(x), which enters the equations of motion of these variables. Thus, any measurement of length, area or volume is, in reality, a measurement of features of the gravitational field.

But what is really formidable in GTR, the truly momentous novelty, is the third point: the Einstein equations, as well as all other equations of physics appropriately modified according to GTR instructions, are fully relational in the Aristotelian-Cartesian sense. This point is independent from the previous one. Let me give first a conceptual, then a technical account of it.

The point is that the only physically meaningful definition of location that makes physical sense within GTR is relational. GTR describes the world as a set of interacting fields and, possibly, other objects. One of these interacting fields is gμν(x). Motion can be defined only as positioning and displacements of these dynamical objects relative to each other.

To describe the motion of a dynamical object, Newton had to assume that acceleration is absolute, namely it is not relative to this or that other dynamical object. Rather, it is relative to a background space. Faraday, Maxwell and Einstein extended the notion of “dynamical object”: the stuff of the world is fields, not just bodies. Finally, GTR tells us that the background space is itself one of these fields. Thus, the circle is closed, and we are back to relationalism: Newton’s motion with respect to space is indeed motion with respect to a dynamical object: the gravitational field.

All this is coded in the active diffeomorphism invariance (diff invariance) of GR. Active diff invariance should not be confused with passive diff invariance, or invariance under change of coordinates. GTR can be formulated in a coordinate free manner, where there are no coordinates, and no changes of coordinates. In this formulation, there field equations are still invariant under active diffs. Passive diff invariance is a property of a formulation of a dynamical theory, while active diff invariance is a property of the dynamical theory itself. A field theory is formulated in manner invariant under passive diffs (or change of coordinates), if we can change the coordinates of the manifold, re-express all the geometric quantities (dynamical and non-dynamical) in the new coordinates, and the form of the equations of motion does not change. A theory is invariant under active diffs, when a smooth displacement of the dynamical fields (the dynamical fields alone) over the manifold, sends solutions of the equations of motion into solutions of the equations of motion. Distinguishing a truly dynamical field, namely a field with independent degrees of freedom, from a nondynamical filed disguised as dynamical (such as a metric field g with the equations of motion Riemann[g]=0) might require a detailed analysis (for instance, Hamiltonian) of the theory. Because active diff invariance is a gauge, the physical content of GTR is expressed only by those quantities, derived from the basic dynamical variables, which are fully independent from the points of the manifold.

In introducing the background stage, Newton introduced two structures: a spacetime manifold, and its non-dynamical metric structure. GTR gets rid of the non-dynamical metric, by replacing it with the gravitational field. More importantly, it gets rid of the manifold, by means of active diff invariance. In GTR, the objects of which the world is made do not live over a stage and do not live on spacetime: they live, so to say, over each other’s shoulders.

Of course, nothing prevents us, if we wish to do so, from singling out the gravitational field as “the more equal among equals”, and declaring that location is absolute in GTR, because it can be defined with respect to it. But this can be done within any relationalism: we can always single out a set of objects, and declare them as not-moving by definition. The problem with this attitude is that it fully misses the great Einsteinian insight: that Newtonian spacetime is just one field among the others. More seriously, this attitude sends us into a nightmare when we have to deal with the motion of the gravitational field itself (which certainly “moves”: we are spending millions for constructing gravity wave detectors to detect its tiny vibrations). There is no absolute referent of motion in GTR: the dynamical fields “move” with respect to each other.

Notice that the third step was not easy for Einstein, and came later than the previous two. Having well understood the first two, but still missing the third, Einstein actively searched for non-generally covariant equations of motion for the gravitational field between 1912 and 1915. With his famous “hole argument” he had convinced himself that generally covariant equations of motion (and therefore, in this context, active diffeomorphism invariance) would imply a truly dramatic revolution with respect to the Newtonian notions of space and time. In 1912 he was not able to take this profoundly revolutionary step, but in 1915 he took this step, and found what Landau calls “the most beautiful of the physical theories”.

# Gauge Geometry and Philosophical Dynamism Weyl was dissatisfied with his own theory of the predicative construction of the arithmetically definable subset of the classical real continuum by the time he had finished his Das Kontinuum, when he compared it with Husserl’s continuum of time, which possessed a “non-atomistic” character in contradistinction with his own theory. No determined point of time could be exhibited, only approximate fixing is possible, just as in the case of “continuum of spatial intuition”. He himself accepted the necessity that the mathematical concept of continuum, the continuous manifold, should not be characterized in terms of modern set theory enriched by topological axioms, because this would contradict the essence of continuum. Weyl says,

It seems that set theory violates against the essence of continuum, which, by its very nature, cannot at all be battered into a single set of elements. not the relationship of an element to a set, but a part of the whole ought to be taken as a basis for the analysis of the continuum.

For Weyl, single points of continuum were empty abstractions, and made him enter a difficult terrain, as no mathematical conceptual frame was in sight, which could satisfy his methodological postulate in a sufficiently elaborative manner. For some years, he sympathized with Brouwer’s idea to characterize points in the intuitionistic one-dimensional continuum by “free choice sequences” of nested intervals, and even tried to extend the idea to higher dimensions and explored the possibility of a purely combinatorial approach to the concept of manifold, in which point-like localizations were given only by infinite sequences of nested star neighborhoods in barycentric subdivisions of a combinatorially defined “manifold”. There arose, however, the problem of how to characterize the local manifold property in purely combinatorial terms.

Weyl was much more successful on another level to rebuild differential geometry in manifolds from a “purely infinitesimal” point of view. He generalized Riemann’s proposal for a differential geometric metric

ds2(x) = ∑n i, j = 1 gij(x) dxi dxj

From his purely infinitesimal point of view, it seemed a strange effect that the length of two vectors ξ(x) and η(x’) given at different points x and x’ can be immediately and objectively compared in this framework after the calculation of

|ξ(x)|2 = ∑n i, j = 1 gij(x) ξi ξj,

|η(x’)|2 = ∑n i, j = 1 gij(x’) ηi ηj

In this context, it was, comparatively easy for Weyl, to give a perfectly infinitesimal characterization of metrical concepts. He started from a well-known structure of conformal metric, i.e. an equivalence class [g] of semi-Riemannian metrics g = gij(x) and g’ = g’ij(x), which are equal up to a point of dependent positive factor λ(x) > 0, g’ = λg. Then, comparison of length made immediate sense only for vectors attached to the same point x, independently of the gauge of the metric, i.e. the choice of the representative in the conformal class. To achieve comparability of lengths of vectors inside each infinitesimal neighborhood, Weyl introduced the conception of length connection formed in analogy to the affine connection, Γ, just distilled from the classical Christoffel Symbols Γkij of Riemannian geometry by Levi Civita. The localization inside such an infinitesimal neighborhood was given, as would have been done already by the mathematicians of the past, by coordinate parameters x and x’ = x + dx for some infinitesimal displacement dx. Weyl’s length connection consisted, then, in an equivalence class of differential I-forms [Ψ], Ψ ≡ ∑ni = 1 Ψidxi, where an equivalent representation of the form is given by Ψ’ ≡ Ψ – d log λ, corresponding to a change of gauge of the conformal metric by the factor λ. Weyl called this transformation, which he recognized as necessary for the consistency of his extended symbol system, the gauge transformation of the length connection.

Weyl established a purely infinitesimal gauge geometry, where lengths of vectors (or derived metrical concepts in tensor fields) were immediately comparable only in the infinitesimal neighborhood of one point, and for points of finite distance only after an integration procedure. this integration turned out to be, in general, path dependent. Independence of the choice of path between two points x and x’ holds if and only if the length curvature vanishes. the concept of curvature was built in direct analogy to the curvature of the affine connection and turned out to be, in this case, just the exterior derivative of the length connection f ≡ dΨ. This led Weyl to a coherent and conceptually pleasing realization of a metrical differential geometry built upon purely infinitesimal principles. moreover, Weyl was convinced of important consequences of his new gauge geometry for physics. The infinitesimal neighborhoods understood as spheres of activity as Fichte might have said, suggested looking for interpretations of the length connection as a field representing physically active quantities. In fact, building on the mathematically obvious observation df ≡ 0, which was formally identical with the second system of the generally covariant Maxwell equations, Weyl immediately drew the conclusion that the length curvature f ought to be identified with the electromagnetic field.

He, however, gave up the belief in the ontological correctness of the purely field-theoretic approach to matter, where the Mie-Hilbert theory of a combined Lagrange function L(g,Ψ) for the action of the gravitational field (g) and electromagnetism (Ψ) was further geometrized and technically enriched by the principle of gauge invariance (L), substituting in its place a philosophically motivated a priori argumentation for the conceptual superiority of his gauge geometry. The goal of a unified description of gravitation and electromagnetism, and the derivation of matter structures from it, was nothing specific to Weyl. In his theory, the purely infinitesimal approach to manifolds and the ensuing possibility to geometrically unify the two-known interaction fields gravitation and electromagnetism took on a dense and conceptually sophisticated form.

# Tensoring Heterotic Superstring Theories The left-moving and right-moving modes of a string can be separated and treated as different theories. In 1984 it was realized that consistent string theories could be built by combining a bosonic string theory moving in one direction along the string, with a supersymmetric string theory with a single q1 moving in the opposite direction. These theories are called heterotic superstring theories. That sounds crazy — because bosonic strings live in 26 dimensions but supersymmetric string theories live in 10 dimensions. But the extra 16 dimensions of the bosonic side of the theory aren’t really spacetime dimensions. Heterotic string theories are supersymmetric string theories living in ten spacetime dimensions. Heterotic string theories are built by tensoring a left- and a right-  moving string which do not have the same base fields. More explicitly, it is constructed by tensoring the right-moving super string with 10 left-moving bosonic and 32 internal left-moving fermionic fields. Internal here means that the field does not transform under Lorentz-transformations, this implies that the boundary conditions on these allow for rotations. The two types of heterotic theories that are possible come from the two types of gauge symmetry that give rise to quantum mechanically consistent theories. The first is SO(32) and the second is the more exotic combination called E8XE8. The E8XE8 heterotic theory was previously regarded as the only string theory that could give realistic physics, until the mid-1990s, when additional possibilities based on the other theories were identified.

# von Neumann & Dis/belief in Hilbert Spaces

I would like to make a confession which may seem immoral: I do not believe absolutely in Hilbert space any more.

— John von Neumann, letter to Garrett Birkhoff, 1935. The mathematics: Let us consider the raison d’ˆetre for the Hilbert space formalism. So why would one need all this ‘Hilbert space stuff, i.e. the continuum structure, the field structure of complex numbers, a vector space over it, inner-product structure, etc. Why? According to von Neumann, he simply used it because it happened to be ‘available’. The use of linear algebra and complex numbers in so many different scientific areas, as well as results in model theory, clearly show that quite a bit of modeling can be done using Hilbert spaces. On the other hand, we can also model any movie by means of the data stream that runs through your cables when watching it. But does this mean that these data streams make up the stuff that makes a movie? Clearly not, we should rather turn our attention to the stuff that is being taught at drama schools and directing schools. Similarly, von Neumann turned his attention to the actual physical concepts behind quantum theory, more specifically, the notion of a physical property and the structure imposed on these by the peculiar nature of quantum observation. His quantum logic gave the resulting ‘algebra of physical properties’ a privileged role. All of this leads us to … the physics of it. Birkhoff and von Neumann crafted quantum logic in order to emphasize the notion of quantum superposition. In terms of states of a physical system and properties of that system, superposition means that the strongest property which is true for two distinct states is also true for states other than the two given ones. In order-theoretic terms this means, representing states by the atoms of a lattice of properties, that the join p ∨ q of two atoms p and q is also above other atoms. From this it easily follows that the distributive law breaks down: given atom r ≠ p, q with r < p ∨ q we have r ∧ (p ∨ q) = r while (r ∧ p) ∨ (r ∧ q) = 0 ∨ 0 = 0. Birkhoff and von Neumann as well as many others believed that understanding the deep structure of superposition is the key to obtaining a better understanding of quantum theory as a whole.

For Schrödinger, this is the behavior of compound quantum systems, described by the tensor product. While the quantum information endeavor is to a great extend the result of exploiting this important insight, the language of the field is still very much that of strings of complex numbers, which is akin to the strings of 0’s and 1’s in the early days of computer programming. If the manner in which we describe compound quantum systems captures so much of the essence of quantum theory, then it should be at the forefront of the presentation of the theory, and not preceded by continuum structure, field of complex numbers, vector space over the latter, etc, to only then pop up as some secondary construct. How much quantum phenomena can be derived from ‘compoundness + epsilon’. It turned out that epsilon can be taken to be ‘very little’, surely not involving anything like continuum, fields, vector spaces, but merely a ‘2D space’ of temporal composition and compoundness, together with some very natural purely operational assertion, including one which in a constructive manner asserts entanglement; among many other things, trace structure then follows.