Network Theoretic of the Fermionic Quantum State – Epistemological Rumination. Thought of the Day 150.0

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In quantum physics, fundamental particles are believed to be of two types: fermions or bosons, depending on the value of their spin (an intrinsic ‘angular moment’ of the particle). Fermions have half-integer spin and cannot occupy a quantum state (a configuration with specified microscopic degrees of freedom, or quantum numbers) that is already occupied. In other words, at most one fermion at a time can occupy one quantum state. The resulting probability that a quantum state is occupied is known as the Fermi-Dirac statistics.

Now, if we want to convert this into a model with maximum entropy, where the real movement is defined topologically, then we require a reproduction of heterogeneity that is observed. The starting recourse is network theory with an ensemble of networks where each vertex i has the same degree ki as in the real network. This choice is justified by the fact that, being an entirely local topological property, the degree is expected to be directly affected by some intrinsic (non-topological) property of vertices. The caveat is that the real shouldn’t be compared with the randomized, which could otherwise lead to interpreting the observed as ‘unavoidable’ topological constraints, in the sense that the violation of the observed values would lead to an ‘impossible’, or at least very unrealistic values.

The resulting model is known as the Configuration Model, and is defined as a maximum-entropy ensemble of graphs with given degree sequence. The degree sequence, which is the constraint defining the model, is nothing but the ordered vector k of degrees of all vertices (where the ith component ki is the degree of vertex i). The ordering preserves the ‘identity’ of vertices: in the resulting network ensemble, the expected degree ⟨ki⟩ of each vertex i is the same as the empirical value ki for that vertex. In the Configuration Model, the graph probability is given by

P(A) = ∏i<jqij(aij) =  ∏i<jpijaij (1 – pij)1-aij —– (1)

where qij(a) = pija (1 – pij)1-a is the probability that particular entry of the adjacency matrix A takes the value aij = a, which is a Bernoulli process with different pairs of vertices characterized by different connection probabilities pij. A Bernoulli trial (or Bernoulli process) is the simplest random event, i.e. one characterized by only two possible outcomes. One of the two outcomes is referred to as the ‘success’ and is assigned a probability p. The other outcome is referred to as the ‘failure’, and is assigned the complementary probability 1 − p. These probabilities read

⟨aij⟩ = pij = (xixj)/(1 + xixj) —– (2)

where xi is the Lagrange multiplier obtained by ensuring that the expected degree of the corresponding vertex i equals its observed value: ⟨ki⟩ = ki ∀ i. As always happens in maximum-entropy ensembles, the probabilistic nature of configurations implies that the constraints are valid only on average (the angular brackets indicate an average over the ensemble of realizable networks). Also note that pij is a monotonically increasing function of xi and xj. This implies that ⟨ki⟩ is a monotonically increasing function of xi. An important consequence is that two variables i and j with the same degree ki = kj must have the same value xi = xj.

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(2) provides an interesting connection with quantum physics, and in particular the statistical mechanics of fermions. The ‘selection rules’ of fermions dictate that only one particle at a time can occupy a single-particle state, exactly as each pair of vertices in binary networks can be either connected or disconnected. In this analogy, every pair i, j of vertices is a ‘quantum state’ identified by the ‘quantum numbers’ i and j. So each link of a binary network is like a fermion that can be in one of the available states, provided that no two objects are in the same state. (2) indicates the expected number of particles/links in the state specified by i and j. With no surprise, it has the same form of the so-called Fermi-Dirac statistics describing the expected number of fermions in a given quantum state. The probabilistic nature of links allows also for the presence of empty states, whose occurrence is now regulated by the probability coefficients (1 − pij). The Configuration Model allows the whole degree sequence of the observed network to be preserved (on average), while randomizing other (unconstrained) network properties. now, when one compares the higher-order (unconstrained) observed topological properties with their expected values calculated over the maximum-entropy ensemble, it should be indicative of the fact that the degree of sequence is informative in explaining the rest of the topology, which is a consequent via probabilities in (2). Colliding these into a scatter plot, the agreement between model and observations can be simply assessed as follows: the less scattered the cloud of points around the identity function, the better the agreement between model and reality. In principle, a broadly scattered cloud around the identity function would indicate the little effectiveness of the chosen constraints in reproducing the unconstrained properties, signaling the presence of genuine higher-order patterns of self-organization, not simply explainable in terms of the degree sequence alone. Thus, the ‘fermionic’ character of the binary model is the mere result of the restriction that no two binary links can be placed between any two vertices, leading to a mathematical result which is formally equivalent to the one of quantum statistics.

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Symmetrical – Asymmetrical Dialectics Within Catastrophical Dynamics. Thought of the Day 148.0

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Catastrophe theory has been developed as a deterministic theory for systems that may respond to continuous changes in control variables by a discontinuous change from one equilibrium state to another. A key idea is that system under study is driven towards an equilibrium state. The behavior of the dynamical systems under study is completely determined by a so-called potential function, which depends on behavioral and control variables. The behavioral, or state variable describes the state of the system, while control variables determine the behavior of the system. The dynamics under catastrophe models can become extremely complex, and according to the classification theory of Thom, there are seven different families based on the number of control and dependent variables.

Let us suppose that the process yt evolves over t = 1,…, T as

dyt = -dV(yt; α, β)dt/dyt —– (1)

where V (yt; α, β) is the potential function describing the dynamics of the state variable ycontrolled by parameters α and β determining the system. When the right-hand side of (1) equals zero, −dV (yt; α, β)/dyt = 0, the system is in equilibrium. If the system is at a non-equilibrium point, it will move back to its equilibrium where the potential function takes the minimum values with respect to yt. While the concept of potential function is very general, i.e. it can be quadratic yielding equilibrium of a simple flat response surface, one of the most applied potential functions in behavioral sciences, a cusp potential function is defined as

−V(yt; α, β) = −1/4yt4 + 1/2βyt2 + αyt —– (2)

with equilibria at

-dV(yt; α, β)dt/dyt = -yt3 + βyt + α —– (3)

being equal to zero. The two dimensions of the control space, α and β, further depend on realizations from i = 1 . . . , n independent variables xi,t. Thus it is convenient to think about them as functions

αx = α01x1,t +…+ αnxn,t —– (4)

βx = β0 + β1x1,t +…+ βnxn,t —– (5)

The control functions αx and βx are called normal and splitting factors, or asymmetry and bifurcation factors, respectively and they determine the predicted values of yt given xi,t. This means that for each combination of values of independent variables there might be up to three predicted values of the state variable given by roots of

-dV(yt; αx, βx)dt/dyt = -yt3 + βyt + α = 0 —– (6)

This equation has one solution if

δx = 1/4αx2 − 1/27βx3 —– (7)

is greater than zero, δx > 0 and three solutions if δx < 0. This construction can serve as a statistic for bimodality, one of the catastrophe flags. The set of values for which the discriminant is equal to zero, δx = 0 is the bifurcation set which determines the set of singularity points in the system. In the case of three roots, the central root is called an “anti-prediction” and is least probable state of the system. Inside the bifurcation, when δx < 0, the surface predicts two possible values of the state variable which means that the state variable is bimodal in this case.

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Most of the systems in behavioral sciences are subject to noise stemming from measurement errors or inherent stochastic nature of the system under study. Thus for a real-world applications, it is necessary to add non-deterministic behavior into the system. As catastrophe theory has primarily been developed to describe deterministic systems, it may not be obvious how to extend the theory to stochastic systems. An important bridge has been provided by the Itô stochastic differential equations to establish a link between the potential function of a deterministic catastrophe system and the stationary probability density function of the corresponding stochastic process. Adding a stochastic Gaussian white noise term to the system

dyt = -dV(yt; αx, βx)dt/dyt + σytdWt —– (8)

where -dV(yt; αx, βx)dt/dyt is the deterministic term, or drift function representing the equilibrium state of the cusp catastrophe, σyt is the diffusion function and Wt is a Wiener process. When the diffusion function is constant, σyt = σ, and the current measurement scale is not to be nonlinearly transformed, the stochastic potential function is proportional to deterministic potential function and probability distribution function corresponding to the solution from (8) converges to a probability distribution function of a limiting stationary stochastic process as dynamics of yt are assumed to be much faster than changes in xi,t. The probability density that describes the distribution of the system’s states at any t is then

fs(y|x) = ψ exp((−1/4)y4 + (βx/2)y2 + αxy)/σ —– (9)

The constant ψ normalizes the probability distribution function so its integral over the entire range equals to one. As bifurcation factor βx changes from negative to positive, the fs(y|x) changes its shape from unimodal to bimodal. On the other hand, αx causes asymmetry in fs(y|x).

The Natural Theoretic of Electromagnetism. Thought of the Day 147.0

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In Maxwell’s theory, the field strength F = 1/2Fμν dxμ ∧ dxν is a real 2-form on spacetime, and thence a natural object at the same time. The homogeneous Maxwell equation dF = 0 is an equation involving forms and it has a well-known local solution F = dA’, i.e. there exists a local spacetime 1-form A’ which is a potential for the field strength F. Of course, if spacetime is contractible, as e.g. for Minkowski space, the solution is also a global one. As is well-known, in the non-commutative Yang-Mills theory case the field strength F = 1/2FAμν TA ⊗ dxμ ∧ dxν is no longer a spacetime form. This is a somewhat trivial remark since the transformation laws of such field strength are obtained as the transformation laws of the curvature of a principal connection with values in the Lie algebra of some (semisimple) non-Abelian Lie group G (e.g. G = SU(n), n 2 ≥ 2). However, the common belief that electromagnetism is to be intended as the particular case (for G =U(1)) of a non-commutative theory is not really physically evident. Even if we subscribe this common belief, which is motivated also by the tremendous success of the quantized theory, let us for a while discuss electromagnetism as a standalone theory.

From a mathematical viewpoint this is a (different) approach to electromagnetism and the choice between the two can be dealt with on a physical ground only. Of course the 1-form A’ is defined modulo a closed form, i.e. locally A” = A’ + dα is another solution.

How can one decide whether the potential of electromagnetism should be considered as a 1-form or rather as a principal connection on a U(1)-bundle? First of all we notice that by a standard hole argument (one can easily define compact supported closed 1-forms, e.g. by choosing the differential of compact supported functions which always exist on a paracompact manifold) the potentials A and A’ represent the same physical situation. On the other hand, from a mathematical viewpoint we would like the dynamical field, i.e. the potential A’, to be a global section of some suitable configuration bundle. This requirement is a mathematical one, motivated on the wish of a well-defined geometrical perspective based on global Variational Calculus.

The first mathematical way out is to restrict attention to contractible spacetimes, where A’ may be always chosen to be global. Then one can require the gauge transformations A” = A’ + dα to be Lagrangian symmetries. In this way, field equations select a whole equivalence class of gauge-equivalent potentials, a procedure which solves the hole argument problem. In this picture the potential A’ is really a 1-form, which can be dragged along spacetime diffeomorphism and which admits the ordinary Lie derivatives of 1-forms. Unfortunately, the restriction to contractible spacetimes is physically unmotivated and probably wrong.

Alternatively, one can restrict electromagnetic fields F, deciding that only exact 2-forms F are allowed. That actually restricts the observable physical situations, by changing the homogeneous Maxwell equations (i.e. Bianchi identities) by requiring that F is not only closed but exact. One should in principle be able to empirically reject this option.

On non-contractible spacetimes, one is necessarily forced to resort to a more “democratic” attitude. The spacetime is covered by a number of patches Uα. On each patch Uα one defines a potential A(α). In the intersection of two patches the two potentials A(α) and A(β) may not agree. In each patch, in fact, the observer chooses his own conventions and he finds a different representative of the electromagnetic potential, which is related by a gauge transformation to the representatives chosen in the neighbour patch(es). Thence we have a family of gauge transformations, one in each intersection Uαβ, which obey cocycle identities. If one recognizes in them the action of U(1) then one can build a principal bundle P = (P, M, π; U(1)) and interpret the ensuing potential as a connection on P. This leads way to the gauge natural formalism.

Anyway this does not close the matter. One can investigate if and when the principal bundle P, in addition to the obvious principal structure, can be also endowed with a natural structure. If that were possible then the bundle of connections Cp (which is associated to P) would also be natural. The problem of deciding whether a given gauge natural bundle can be endowed with a natural structure is quite difficult in general and no full theory is yet completely developed in mathematical terms. That is to say, there is no complete classification of the topological and differential geometric conditions which a principal bundle P has to satisfy in order to ensure that, among the principal trivializations which determine its gauge natural structure, one can choose a sub-class of trivializations which induce a purely natural bundle structure. Nor it is clear how many inequivalent natural structures a good principal bundle may support. Though, there are important examples of bundles which support at the same time a natural and a gauge natural structure. Actually any natural bundle is associated to some frame bundle L(M), which is principal; thence each natural bundle is also gauge natural in a trivial way. Since on any paracompact manifold one can choose a global Riemannian metric g, the corresponding tangent bundle T(M) can be associated to the orthonormal frame bundle O(M, g) besides being obviously associated to L(M). Thence the natural bundle T(M) may be also endowed with a gauge natural bundle structure with structure group O(m). And if M is orientable the structure can be further reduced to a gauge natural bundle with structure group SO(m).

Roughly speaking, the task is achieved by imposing restrictions to cocycles which generate T(M) according to the prescription by imposing a privileged class of changes of local laboratories and sets of measures. Imposing the cocycle ψ(αβ) to take its values in O(m) rather than in the larger group GL(m). Inequivalent gauge natural structures are in one-to-one correspondence with (non isometric) Riemannian metrics on M. Actually whenever there is a Lie group homomorphism ρ : GU(m) → G for some s onto some given Lie group G we can build a natural G-principal bundle on M. In fact, let (Uα, ψ(α)) be an atlas of the given manifold M, ψ(αβ) be its transition functions and jψ(αβ) be the induced transition functions of L(M). Then we can define a G-valued cocycle on M by setting ρ(jψ(αβ)) and thence a (unique up to fibered isomorphisms) G-principal bundle P(M) = (P(M), M, π; G). The bundle P(M), as well as any gauge natural bundle associated to it, is natural by construction. Now, defining a whole family of natural U(1)-bundles Pq(M) by using the bundle homomorphisms

ρq: GL(m) → U(1): J ↦ exp(iq ln det|J|) —– (1)

where q is any real number and In denotes the natural logarithm. In the case q = 0 the image of ρ0 is the trivial group {I}; and, all the induced bundles are trivial, i.e. P = M x U(1).

The natural lift φ’ of a diffeomorphism φ: M → M is given by

φ'[x, e]α = [φ(x), eiq ln det|J|. e]α —– (2)

where J is the Jacobin of the morphism φ. The bundles Pq(M) are all trivial since they allow a global section. In fact, on any manifold M, one can define a global Riemannian metric g, where the local sections glue together.

Since the bundles Pq(M) are all trivial, they are all isomorphic to M x U(1) as principal U(1)-bundles, though in a non-canonical way unless q = 0. Any two of the bundles Pq1(M) and Pq2(M) for two different values of q are isomorphic as principal bundles but the isomorphism obtained is not the lift of a spacetime diffeomorphism because of the two different values of q. Thence they are not isomorphic as natural bundles. We are thence facing a very interesting situation: a gauge natural bundle C associated to the trivial principal bundle P can be endowed with an infinite family of natural structures, one for each q ∈ R; each of these natural structures can be used to regard principal connections on P as natural objects on M and thence one can regard electromagnetism as a natural theory.

Now that the mathematical situation has been a little bit clarified, it is again a matter of physical interpretation. One can in fact restrict to electromagnetic potentials which are a priori connections on a trivial structure bundle P ≅ M x U(1) or to accept that more complicated situations may occur in Nature. But, non-trivial situations are still empirically unsupported, at least at a fundamental level.

Gauge Fixity Towards Hyperbolicity: The Case For Equivalences. Part 2.

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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)

Gauge Fixity Towards Hyperbolicity: General Theory of Relativity and Superpotentials. Part 1.

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Gravitational field is described by a pseudo-Riemannian metric g (with Lorentzian signature (1, m-1)) over the spacetime M of dimension dim(M) = m; in standard General Relativity, m = 4. The configuration bundle is thence the bundle of Lorentzian metrics over M, denoted by Lor(M) . The Lagrangian is second order and it is usually chosen to be the so-called Hilbert Lagrangian:

LH: J2Lor(m) → ∧om(M)

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

where

R = gαβ Rαβ denotes the scalar curvature, √g the square root of the absolute value of the metric determinant and ∧ is a real constant (called the cosmological constant). The coupling constant 1/2κ which is completely irrelevant until the gravitational field is not coupled to some other field, depends on conventions; in natural units, i.e. c = 1, h = 1, G = 1, dimension 4 and signature ( + , – , – , – ) one has κ = – 8π.

Field equations are the well known Einstein equations with cosmological constant

Rαβ – 1/2 Rgαβ = -∧gαβ —— (2)

Lagrangian momenta is defined by:

pαβ = ∂LH/∂gαβ = 1/2κ (Rαβ – 1/2(R – 2∧)gαβ)√g

Pαβ = ∂LH/∂Rαβ = 1/2κ gαβ√g —– (3)

Thus the covariance identity is the following:

dα(LHξα) = pαβ£ξgαβ + Pαβ£ξRαβ —– (4)

or equivalently,

α(LHξα) = pαβ£ξgαβ + PαβεξΓεαβ – δεβ£ξΓλαλ) —– (5)

where ∇ε denotes the covariant derivative with respect to the Levi-Civita connection of g. Thence we have a weak conservation law for the Hilbert Lagrangian

Div ε(LH, ξ) = W(LH, ξ) —– (6)

Conserved currents and work forms have respectively the following expressions:

ε(LH, ξ) = [Pαβ£ξΓεαβ – Pαε£ξΓλαλ – LHξε]dsε = √g/2κ(gαβgεσ – gσβgεα) ∇α£ξgβσdsε – √g/2κξεRdsε = √g/2κ[(3/2Rαλ – (R – 2∧)δαλλ + (gβγδαλ – gα(γδβ)λβγξλ]dsα —– (7)

W(LH, ξ) = √g/κ(Rαβ – 1/2(R – 2∧)gαβ)∇(αξβ)ds —– (8)

As any other natural theory, General Relativity allows superpotentials. In fact, the current can be recast into the form:

ε(LH, ξ) = ε'(LH, ξ) + Div U(LH, ξ) —– (9)

where we set

ε'(LH, ξ) = √g/κ(Rαβ – 1/2(R – 2∧)δαββ)dsα

U(LH, ξ) = 1/2κ ∇[βξα] √gdsαβ —– (10)

The superpotential (10) generalizes to an arbitrary vector field ξ, the well known Komar superpotential which is originally derived for timelike Killing vectors. Whenever spacetime is assumed to be asymptotically fiat, then the superpotential of Komar is known to produce upon integration at spatial infinity ∞ the correct value for angular momentum (e.g. for Kerr-Newman solutions) but just one half of the expected value of the mass. The classical prescriptions are in fact:

m = 2∫ U(LH, ∂t, g)

J = ∫ U(LH, ∂φ, g) —– (11)

For an asymptotically flat solution (e.g. the Kerr-Newman black hole solution) m coincides with the so-called ADM mass and J is the so-called (ADM) angular momentum. For the Kerr-Newman solution in polar coordinates (t, r, θ, φ) the vector fields ∂t and ∂φ are the Killing vectors which generate stationarity and axial symmetry, respectively. Thence, according to this prescription, U(LH, ∂φ) is the superpotential for J while 2U(LH, ∂t) is the superpotential for m. This is known as the anomalous factor problem for the Komar potential. To obtain the expected values for all conserved quantities from the same superpotential, one has to correct the superpotential (10) by some ad hoc additional boundary term. Equivalently and alternatively, one can deduce a corrected superpotential as the canonical superpotential for a corrected Lagrangian, which is in fact the first order Lagrangian for standard General Relativity. This can be done covariantly, provided that one introduces an extra connection Γ’αβμ. The need of a reference connection Γ’ should be also motivated by physical considerations, according to which the conserved quantities have no absolute meaning but they are intrinsically relative to an arbitrarily fixed vacuum level. The simplest choice consists, in fact, in fixing a background metric g (not necessarily of the correct Lorentzian signature) and assuming Γ’ to be the Levi-Civita connection of g. This is rather similar to the gauge fixing à la Hawking which allows to show that Einstein equations form in fact an essentially hyperbolic PDE system. Nothing prevents, however, from taking Γ’ to be any (in principle torsionless) connection on spacetime; also this corresponds to a gauge fixing towards hyperbolicity.

Now, using the term background for a field which enters a field theory in the same way as the metric enters Yang-Mills theory, we see that the background has to be fixed once for all and thence preserved, e.g. by symmetries and deformations. A background has no field equations since deformations fix it; it eventually destroys the naturality of a theory, since fixing the background results in allowing a smaller group of symmetries G ⊂ Diff(M). Accordingly, in truly natural field theories one should not consider background fields either if they are endowed with a physical meaning (as the metric in Yang-Mills theory does) or if they are not.

On the contrary we shall use the expression reference or reference background to denote an extra dynamical field which is not endowed with a direct physical meaning. As long as variational calculus is concerned, reference backgrounds behave in exactly the same way as other dynamical fields do. They obey field equations and they can be dragged along deformations and symmetries. It is important to stress that such a behavior has nothing to do with a direct physical meaning: even if a reference background obeys field equations this does not mean that it is observable, i.e. it can be measured in a laboratory. Of course, not any dynamical field can be treated as a reference background in the above sense. 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….

The Canonical of a priori and a posteriori Variational Calculus as Phenomenologically Driven. Note Quote.

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The expression variational calculus usually identifies two different but related branches in Mathematics. The first aimed to produce theorems on the existence of solutions of (partial or ordinary) differential equations generated by a variational principle and it is a branch of local analysis (usually in Rn); the second uses techniques of differential geometry to deal with the so-called variational calculus on manifolds.

The local-analytic paradigm is often aimed to deal with particular situations, when it is necessary to pay attention to the exact definition of the functional space which needs to be considered. That functional space is very sensitive to boundary conditions. Moreover, minimal requirements on data are investigated in order to allow the existence of (weak) solutions of the equations.

On the contrary, the global-geometric paradigm investigates the minimal structures which allow to pose the variational problems on manifolds, extending what is done in Rn but usually being quite generous about regularity hypotheses (e.g. hardly ever one considers less than C-objects). Since, even on manifolds, the search for solutions starts with a local problem (for which one can use local analysis) the global-geometric paradigm hardly ever deals with exact solutions, unless the global geometric structure of the manifold strongly constrains the existence of solutions.

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A further a priori different approach is the one of Physics. In Physics one usually has field equations which are locally given on a portion of an unknown manifold. One thence starts to solve field equations locally in order to find a local solution and only afterwards one tries to find the maximal analytical extension (if any) of that local solution. The maximal extension can be regarded as a global solution on a suitable manifold M, in the sense that the extension defines M as well. In fact, one first proceeds to solve field equations in a coordinate neighbourhood; afterwards, one changes coordinates and tries to extend the found solution out of the patches as long as it is possible. The coordinate changes are the cocycle of transition functions with respect to the atlas and they define the base manifold M. This approach is essential to physical applications when the base manifold is a priori unknown, as in General Relativity, and it has to be determined by physical inputs.

Luckily enough, that approach does not disagree with the standard variational calculus approach in which the base manifold M is instead fixed from the very beginning. One can regard the variational problem as the search for a solution on that particular base manifold. Global solutions on other manifolds may be found using other variational principles on different base manifolds. Even for this reason, the variational principle should be universal, i.e. one defines a family of variational principles: one for each base manifold, or at least one for any base manifold in a “reasonably” wide class of manifolds. The strong requirement, which is physically motivated by the belief that Physics should work more or less in the same way regardless of the particular spacetime which is actually realized in Nature. Of course, a scenario would be conceivable in which everything works because of the particular (topological, differentiable, etc.) structure of the spacetime. This position, however, is not desirable from a physical viewpoint since, in this case, one has to explain why that particular spacetime is realized (a priori or a posteriori).

In spite of the aforementioned strong regularity requirements, the spectrum of situations one can encounter is unexpectedly wide, covering the whole of fundamental physics. Moreover, it is surprising how the geometric formalism is effectual for what concerns identifications of basic structures of field theories. In fact, just requiring the theory to be globally well-defined and to depend on physical data only, it often constrains very strongly the choice of the local theories to be globalized. These constraints are one of the strongest motivations in choosing a variational approach in physical applications. Another motivation is a well formulated framework for conserved quantities. A global- geometric framework is a priori necessary to deal with conserved quantities being non-local.

In the modem perspective of Quantum Field Theory (QFT) the basic object encoding the properties of any quantum system is the action functional. From a quantum viewpoint the action functional is more fundamental than field equations which are obtained in the classical limit. The geometric framework provides drastic simplifications of some key issues, such as the definition of the variation operator. The variation is deeply geometric though, in practice, it coincides with the definition given in the local-analytic paradigm. In the latter case, the functional derivative is usually the directional derivative of the action functional which is a function on the infinite-dimensional space of fields defined on a region D together with some boundary conditions on the boundary ∂D. To be able to define it one should first define the functional space, then define some notion of deformation which preserves the boundary conditions (or equivalently topologize the functional space), define a variation operator on the chosen space, and, finally, prove the most commonly used properties of derivatives. Once one has done it, one finds in principle the same results that would be found when using the geometric definition of variation (for which no infinite dimensional space is needed). In fact, in any case of interest for fundamental physics, the functional derivative is simply defined by means of the derivative of a real function of one real variable. The Lagrangian formalism is a shortcut which translates the variation of (infinite dimensional) action functionals into the variation of the (finite dimensional) Lagrangian structure.

Another feature of the geometric framework is the possibility of dealing with non-local properties of field theories. There are, in fact, phenomena, such as monopoles or instantons, which are described by means of non-trivial bundles. Their properties are tightly related to the non-triviality of the configuration bundle; and they are relatively obscure when regarded by any local paradigm. In some sense, a local paradigm hides global properties in the boundary conditions and in the symmetries of the field equations, which are in turn reflected in the functional space we choose and about which, it being infinite dimensional, we do not know almost anything a priori. We could say that the existence of these phenomena is a further hint that field theories have to be stated on bundles rather than on Cartesian products. This statement, if anything, is phenomenologically driven.

When a non-trivial bundle is involved in a field theory, from a physical viewpoint it has to be regarded as an unknown object. As for the base manifold, it has then to be constructed out of physical inputs. One can do that in (at least) two ways which are both actually used in applications. First of all, one can assume the bundle to be a natural bundle which is thence canonically constructed out of its base manifold. Since the base manifold is identified by the (maximal) extension of the local solutions, then the bundle itself is identified too. This approach is the one used in General Relativity. In these applications, bundles are gauge natural and they are therefore constructed out of a structure bundle P, which, usually, contains extra information which is not directly encoded into the spacetime manifolds. In physical applications the structure bundle P has also to be constructed out of physical observables. This can be achieved by using gauge invariance of field equations. In fact, two local solutions differing by a (pure) gauge transformation describe the same physical system. Then while extending from one patch to another we feel free both to change coordinates on M and to perform a (pure) gauge transformation before glueing two local solutions. Then coordinate changes define the base manifold M, while the (pure) gauge transformations form a cocycle (valued in the gauge group) which defines, in fact, the structure bundle P. Once again solutions with different structure bundles can be found in different variational principles. Accordingly, the variational principle should be universal with respect to the structure bundle.

Local results are by no means less important. They are often the foundations on which the geometric framework is based on. More explicitly, Variational Calculus is perhaps the branch of mathematics that possibilizes the strongest interaction between Analysis and Geometry.

The Closed String Cochain Complex C is the String Theory Substitute for the de Rham Complex of Space-Time. Note Quote.

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In closed string theory the central object is the vector space C = CS1 of states of a single parameterized string. This has an integer grading by the “ghost number”, and an operator Q : C → C called the “BRST operator” which raises the ghost number by 1 and satisfies Q2 = 0. In other words, C is a cochain complex. If we think of the string as moving in a space-time M then C is roughly the space of differential forms defined along the orbits of the action of the reparametrization group Diff+(S1) on the free loop space LM (more precisely, square-integrable forms of semi-infinite degree). Similarly, the space C of a topologically-twisted N = 2 supersymmetric theory, is a cochain complex which models the space of semi-infinite differential forms on the loop space of a Kähler manifold – in this case, all square-integrable differential forms, not just those along the orbits of Diff+(S1). In both kinds of example, a cobordism Σ from p circles to q circles gives an operator UΣ,μ : C⊗p → C⊗q which depends on a conformal structure μ on Σ. This operator is a cochain map, but its crucial feature is that changing the conformal structure μ on Σ changes the operator UΣ,μ only by a cochain homotopy. The cohomology H(C) = ker(Q)/im(Q) – the “space of physical states” in conventional string theory – is therefore the state space of a topological field theory.

A good way to describe how the operator UΣ,μ varies with μ is as follows:

If MΣ is the moduli space of conformal structures on the cobordism Σ, modulo diffeomorphisms of Σ which are the identity on the boundary circles, then we have a cochain map

UΣ : C⊗p → Ω(MΣ, C⊗q)

where the right-hand side is the de Rham complex of forms on MΣ with values in C⊗q. The operator UΣ,μ is obtained from UΣ by restricting from MΣ to {μ}. The composition property when two cobordisms Σ1 and Σ2 are concatenated is that the diagram

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commutes, where the lower horizontal arrow is induced by the map MΣ1 × MΣ2 → MΣ2 ◦ Σ1 which expresses concatenation of the conformal structures.

For each pair a, b of boundary conditions we shall still have a vector space – indeed a cochain complex – Oab, but it is no longer the space of morphisms from b to a in a category. Rather, what we have is an A-category. Briefly, this means that instead of a composition law Oab × Obc → Oac we have a family of ways of composing, parametrized by the contractible space of conformal structures on the surface of the figure:

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In particular, any two choices of a composition law from the family are cochain homotopic. Composition is associative in the sense that we have a contractible family of triple compositions Oab × Obc × Ocd → Oad, which contains all the maps obtained by choosing a binary composition law from the given family and bracketing the triple in either of the two possible ways.

This is not the usual way of defining an A-structure. According to Stasheff’s original definition, an A-structure on a space X consists of a sequence of choices: first, a composition law m2 : X × X → X; then, a choice of a map

m3 : [0, 1] × X × X × X → X which is a homotopy between

(x, y, z) ↦ m2(m2(x, y), z) and (x, y, z) ↦ m2(x, m2(y, z)); then, a choice of a map

m4 : S4 × X4 → X,

where S4 is a convex plane polygon whose vertices are indexed by the five ways of bracketing a 4-fold product, and m4|((∂S4) × X4) is determined by m3; and so on. There is an analogous definition – applying to cochain complexes rather than spaces.

Apart from the composition law, the essential algebraic properties are the non-degenerate inner product, and the commutativity of the closed algebra C. Concerning the latter, when we pass to cochain theories the multiplication in C will of course be commutative up to cochain homotopy, but, the moduli space MΣ of closed string multiplications i.e., the moduli space of conformal structures on a pair of pants Σ, modulo diffeomorphisms of Σ which are the identity on the boundary circles, is not contractible: it has the homotopy type of the space of ways of embedding two copies of the standard disc D2 disjointly in the interior of D2 – this space of embeddings is of course a subspace of MΣ. In particular, it contains a natural circle of multiplications in which one of the embedded discs moves like a planet around the other, and there are two different natural homotopies between the multiplication and the reversed multiplication. This might be a clue to an important difference between stringy and classical space-times. The closed string cochain complex C is the string theory substitute for the de Rham complex of space-time, an algebra whose multiplication is associative and (graded)commutative on the nose. Over the rationals or the real or complex numbers, such cochain algebras model the category of topological spaces up to homotopy, in the sense that to each such algebra C, we can associate a space XC and a homomorphism of cochain algebras from C to the de Rham complex of XC which is a cochain homotopy equivalence. If we do not want to ignore torsion in the homology of spaces we can no longer encode the homotopy type in a strictly commutative cochain algebra. Instead, we must replace commutative algebras with so-called E-algebras, i.e., roughly, cochain complexes C over the integers equipped with a multiplication which is associative and commutative up to given arbitrarily high-order homotopies. An arbitrary space X has an E-algebra CX of cochains, and conversely one can associate a space XC to each E-algebra C. Thus we have a pair of adjoint functors, just as in rational homotopy theory. The cochain algebras of closed string theory have less higher commutativity than do E-algebras, and this may be an indication that we are dealing with non-commutative spaces that fits in well with the interpretation of the B-field of a string background as corresponding to a bundle of matrix algebras on space-time. At the same time, the non-degenerate inner product on C – corresponding to Poincaré duality – seems to show we are concerned with manifolds, rather than more singular spaces.

Let us consider the category K of cochain complexes of finitely generated free abelian groups and cochain homotopy classes of cochain maps. This is called the derived category of the category of finitely generated abelian groups. Passing to cohomology gives us a functor from K to the category of Z-graded finitely generated abelian groups. In fact the subcategory K0 of K consisting of complexes whose cohomology vanishes except in degree 0 is actually equivalent to the category of finitely generated abelian groups. But the category K inherits from the category of finitely generated free abelian groups a duality functor with properties as ideal as one could wish: each object is isomorphic to its double dual, and dualizing preserves exact sequences. (The dual C of a complex C is defined by (C)i = Hom(C−i, Z).) There is no such nice duality in the category of finitely generated abelian groups. Indeed, the subcategory K0 is not closed under duality, for the dual of the complex CA corresponding to a group A has in general two non-vanishing cohomology groups: Hom(A,Z) in degree 0, and in degree +1 the finite group Ext1(A,Z) Pontryagin-dual to the torsion subgroup of A. This follows from the exact sequence:

0 → Hom(A, Z) → Hom(FA, Z) → Hom(RA, Z) → Ext1(A, Z) → 0

derived from an exact sequence

0 → RA → FA → A → 0

The category K also has a tensor product with better properties than the tensor product of abelian groups, and, better still, there is a canonical cochain functor from (locally well-behaved) compact spaces to K which takes Cartesian products to tensor products.