From God’s Perspective, There Are No Fields…Justified Newtonian, Unjustified Relativistic Claim. Note Quote.

Electromagnetism is a relativistic theory. Indeed, it had been relativistic, or Lorentz invariant, before Einstein and Minkowski understood that this somewhat peculiar symmetry of Maxwell’s equations was not accidental but expressive of a radically new structure of time and space. Minkowski spacetime, in contrast to Newtonian spacetime, doesn’t come with a preferred space-like foliation, its geometric structure is not one of ordered slices representing “objective” hyperplanes of absolute simultaneity. But Minkowski spacetime does have an objective (geometric) structure of light-cones, with one double-light-cone originating in every point. The most natural way to define a particle interaction in Minkowski spacetime is to have the particles interact directly, not along equal-time hyperplanes but along light-cones

In other words, if z_i(τ_􏱁i)  and z_j(τ_j􏱁) denote the trajectories of two charged particles, it wouldn’t make sense to say that the particles interact at “equal times” as it is in Newtonian theory. It would however make perfectly sense to say that the particles interact whenever

(z^μ_i z^μ_j)(z^μ_i z^μ_j) = (z_i – z_j)² = 0 —– (1)

For an observer finding himself in a universe guided by such laws it might then seem like the effects of particle interactions were propagating through space with the speed of light. And this observer may thus insist that there must be something in addition to the particles, something moving or evolving in spacetime and mediating interactions between charged particles. And all this would be a completely legitimate way of speaking, only that it would reflect more about how things appear from a local perspective in a particular frame of reference than about what is truly and objectively going on in the physical world. From “Gods perspective” there are no fields (or photons, or anything of that kind) – only particles in spacetime interacting with each other. This might sound hypothetical, but, it actually is not entirely fictitious. for such a formulation of electrodynamics actually exists and is known as Wheeler-Feynman electrodynamics, or Wheeler-Feynman Absorber Theory. There is a formal property of field equations called “gauge invariance” which makes it possible to look at things in several different, but equivalent, ways. Because of gauge invariance, this theory says that when you push on something, it creates a disturbance in the gravitational field that propagates outward into the future. Out there in the distant future the disturbance interacts with chiefly the distant matter in the universe. It wiggles. When it wiggles it sends a gravitational disturbance backward in time (a so-called “advanced” wave). The effect of all of these “advanced” disturbances propagating backward in time is to create the inertial reaction force you experience at the instant you start to push (and cancel the advanced wave that would otherwise be created by you pushing on the object). So, in this view fields do not have a real existence independent of the sources that emit and absorb them. It is defined by the principle of least action.

Wheeler–Feynman electrodynamics and Maxwell–Lorentz electrodynamics are for all practical purposes empirically equivalent, and it may seem that the choice between the two candidate theories is merely one of convenience and philosophical preference. But this is not really the case since the sad truth is that the field theory, despite its phenomenal success in practical applications and the crucial role it played in the development of modern physics, is inconsistent. The reason is quite simple. The Maxwell–Lorentz theory for a system of N charged particles is defined, as it should be, by a set of mathematical equations. The equation of motion for the particles is given by the Lorentz force law, which is

The electromagnetic force F on a test charge at a given point and time is a certain function of its charge q and velocity v, which can be parameterized by exactly two vectors E and B, in the functional form:

describing the acceleration of a charged particle in an electromagnetic field. The electromagnetic field, represented by the field-tensor F^μν, is described by Maxwell’s equations. The homogenous Maxwell equations tell us that the antisymmetric tensor F^μν (a 2-form) can be written as the exterior derivative of a potential (a 1-form) A^μ(x), i.e. as

F^μν = ∂^μ A^ν – ∂^ν A^μ —– (2)

The inhomogeneous Maxwell equations couple the field degrees of freedom to matter, that is, they tell us how the charges determine the configuration of the electromagnetic field. Fixing the gauge-freedom contained in (2) by demanding ∂_μA^μ(x) = 0 (Lorentz gauge), the remaining Maxwell equations take the particularly simple form:

□ A^μ = – 4π j^μ —– (3)

where

□ = ∂_μ∂^μ

is the d’Alembert operator and j^μ the 4-current density.

The light-cone structure of relativistic spacetime is reflected in the Lorentz-invariant equation (3). The Liénard–Wiechert field at spacetime point x depends on the trajectories of the particles at the points of intersection with the (past and future) light-cones originating in x. The Liénard–Wiechert field (the solution of (3)) is singular precisely at the points where it is needed, namely on the world-lines of the particles. This is the notorious problem of the electron self-interaction: a charged particle generates a field, the field acts back on the particle, the field-strength becomes infinite at the point of the particle and the interaction terms blow up. Hence, the simple truth is that the field concept for managing interactions between point-particles doesn’t work, unless one relies on formal manipulations like renormalization or modifies Maxwell’s laws on small scales. However, we don’t need the fields and by taking the idea of a relativistic interaction theory seriously, we can “cut the middle man” and let the particles interact directly. The status of the Maxwell equation’s (3) in Wheeler–Feynman theory is now somewhat analogous to the status of Laplace’s equation in Newtonian gravity. We can get to the Gallilean invariant theory by writing the force as the gradient of a potential and having that potential satisfy the simplest nontrivial Galilean invariant equation, which is the Laplace equation:

∆V(x, t) = ∑_iδ(x – x_i(t)) —– (4)

Similarly, we can get the (arguably) simplest Lorentz invariant theory by writing the force as the exterior derivative of a potential and having that potential satisfy the simplest nontrivial Lorentz invariant equation, which is (3). And as concerns the equation of motion for the particles, the trajectories, if, are parametrized by proper time, then the Minkowski norm of the 4-velocity is a constant of motion. In Newtonian gravity, we can make sense of the gravitational potential at any point in space by conceiving its effect on a hypothetical test particle, feeling the gravitational force without gravitating itself. However, nothing in the theory suggests that we should take the potential seriously in that way and conceive of it as a physical field. Indeed, the gravitational potential is really a function on configuration space rather than a function on physical space, and it is really a useful mathematical tool rather than corresponding to physical degrees of freedom. From the point of view of a direct interaction theory, an analogous reasoning would apply in the relativistic context. It may seem (and historically it has certainly been the usual understanding) that (3), in contrast to (4), is a dynamical equation, describing the temporal evolution of something. However, from a relativistic perspective, this conclusion seems unjustified.

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The Natural Theoretic of Electromagnetism. Thought of the Day 147.0

In Maxwell’s theory, the field strength F = 1/2F_μν dx^μ ∧ dx^ν 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 = 1/2F^A_μν T_A ⊗ dx^μ ∧ dx^ν 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 U_α. On each patch U_α 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 U_αβ, 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 C_p (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 (U_α, 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 P_q(M) by using the bundle homomorphisms

ρ_q: 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^iθ]_α = [φ(x), e^{iq ln det|J|}. e^iθ]_α —– (2)

where J is the Jacobin of the morphism φ. The bundles P_q(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 P_q(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 P_q1(M) and P_q2(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: General Theory of Relativity and Superpotentials. Part 1.

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:

L_H: J²Lor(m) → ∧^o_m(M)

L_H: L_H(g^αβ, R_αβ)ds = 1/2κ (R – 2∧)√g ds —– (1)

where

R = g^αβ R_αβ 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

R_αβ – 1/2 Rg_αβ = -∧g_αβ —— (2)

Lagrangian momenta is defined by:

p_αβ = ∂L_H/∂g^αβ = 1/2κ (R_αβ – 1/2(R – 2∧)g_αβ)√g

P^αβ = ∂L_H/∂R_αβ = 1/2κ g^αβ√g —– (3)

Thus the covariance identity is the following:

d_α(L_Hξ^α) = p_αβ£_ξg^αβ + P^αβ£_ξR_αβ —– (4)

or equivalently,

∇_α(L_Hξ^α) = p_αβ£_ξg^αβ + P^αβ∇_ε(£_ξΓ^ε_αβ – δ^ε_β£_ξΓ^λ_αλ) —– (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(L_H, ξ) = W(L_H, ξ) —– (6)

Conserved currents and work forms have respectively the following expressions:

ε(L_H, ξ) = [P^αβ£_ξΓ^ε_αβ – P^αε£_ξΓ^λ_αλ – L_Hξ^ε]ds_ε = √g/2κ(g^αβg^εσ – g^σβg^εα) ∇_α£_ξg^βσds_ε – √g/2κξ^εRds_ε = √g/2κ[(3/2R^α_λ – (R – 2∧)δ^α_λ)ξ^λ + (g^βγδ^α_λ – g^α(γδ^β)_λ ∇_βγξ^λ]ds_α —– (7)

W(L_H, ξ) = √g/κ(R_αβ – 1/2(R – 2∧)g_αβ)∇(^αξ^β)ds —– (8)

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

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

where we set

ε'(L_H, ξ) = √g/κ(R^α_β – 1/2(R – 2∧)δ^α_β)ξ^β)ds_α

U(L_H, ξ) = 1/2κ ∇[^βξ^α] √gds_αβ —– (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(L_H, ∂_t, g)

J = ∫_∞ U(L_H, ∂_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(L_H, ∂_φ) is the superpotential for J while 2U(L_H, ∂_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’^α_βμ. 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 Closed String Cochain Complex C is the String Theory Substitute for the de Rham Complex of Space-Time. Note Quote.

In closed string theory the central object is the vector space C = C_S¹ 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 U_Σ,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 U_Σ,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 U_Σ,μ varies with μ is as follows:

If M_Σ 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

U_Σ : C^⊗p → Ω^∗(M_Σ, C^⊗q)

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

commutes, where the lower horizontal arrow is induced by the map M_Σ1 × M_Σ2 → M_{Σ2 ◦ 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 – O_ab, 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 O_ab × O_bc → O_ac we have a family of ways of composing, parametrized by the contractible space of conformal structures on the surface of the figure:

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 O_ab × O_bc × O_cd → O_ad, 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 M_Σ 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 M_Σ. 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 X^C 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 C_X of cochains, and conversely one can associate a space X_C 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 C_A 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(F_A, Z) → Hom(R_A, Z) → Ext¹(A, Z) → 0

derived from an exact sequence

0 → R_A → F_A → 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.

Breakdown of Lorentz Invariance: The Order of Quantum Gravity Phenomenology. Thought of the Day 132.0

The purpose of quantum gravity phenomenology is to analyze the physical consequences arising from various models of quantum gravity. One hope for obtaining an experimental grasp on quantum gravity is the generic prediction arising in many (but not all) quantum gravity models that ultraviolet physics at or near the Planck scale, M_Planck = 1.2 × 10¹⁹ GeV/c², (or in some models the string scale), typically induces violations of Lorentz invariance at lower scales. Interestingly most investigations, even if they arise from quite different fundamental physics, seem to converge on the prediction that the breakdown of Lorentz invariance can generically become manifest in the form of modified dispersion relations

ω² = ω₀² + (1 + η₂) c²k² + η₄(ħ/M_{Lorentz violation})² + k⁴ + ….

where the coefficients η_n are dimensionless (and possibly dependent on the particle species under consideration). The particular inertial frame for these dispersion relations is generally specified to be the frame set by cosmological microwave background, and M_{Lorentz violation} is the scale of Lorentz symmetry breaking which furthermore is generally assumed to be of the order of M_Planck.

Although several alternative scenarios have been considered to justify the modified kinematics,the most commonly explored avenue is an effective field theory (EFT) approach. Here, the focus is explicitly on the class of non-renormalizable EFTs with Lorentz violations associated to dispersion relations. Even if this framework as a “test theory” is successful, it is interesting to note that there are still significant open issues concerning its theoretical foundations. Perhaps the most pressing one is the so called naturalness problem which can be expressed in the following way: The lowest-order correction, proportional to η₂, is not explicitly Planck suppressed. This implies that such a term would always be dominant with respect to the higher-order ones and grossly incompatible with observations (given that we have very good constraints on the universality of the speed of light for different elementary particles). If one were to take cues from observational leads, it is assumed either that some symmetry (other than Lorentz invariance) enforces the η₂ coefficients to be exactly zero, or that the presence of some other characteristic EFT mass scale μ ≪ M_Planck (e.g., some particle physics mass scale) associated with the Lorentz symmetry breaking might enter in the lowest order dimensionless coefficient η₂, which will be then generically suppressed by appropriate ratios of this characteristic mass to the Planck mass: η₂ ∝ (μ/M_Planck)^σ where σ ≥ 1 is some positive power (often taken as one or two). If this is the case then one has two distinct regimes: For low momenta p/(M_Planckc) ≪ (μ/M_Planck)^σ the lower-order (quadratic in the momentum) deviations will dominate over the higher-order ones, while at high energies p/(M_Planckc) ≫ (μ/M_Planck)^σ the higher order terms will be dominant.

The naturalness problem arises because such a scenario is not well justified within an EFT framework; in other words there is no natural suppression of the low-order modifications. EFT cannot justify why only the dimensionless coefficients of the n ≤ 2 terms should be suppressed by powers of the small ratio μ/M_Planck. Even worse, renormalization group arguments seem to imply that a similar mass ratio, μ/M_Planck would implicitly be present also in all the dimensionless n > 2 coefficients, hence suppressing them even further, to the point of complete undetectability. Furthermore, without some protecting symmetry, it is generic that radiative corrections due to particle interactions in an EFT with only Lorentz violations of order n > 2 for the free particles, will generate n = 2 Lorentz violating terms in the dispersion relation, which will then be dominant. Naturalness in EFT would then imply that the higher order terms are at least as suppressed as this, and hence beyond observational reach.

A second issue is that of universality, which is not so much a problem, as an issue of debate as to the best strategy to adopt. In dealing with situations with multiple particles one has to choose between the case of universal (particle-independent) Lorentz violating coefficients η_n, or instead go for a more general ansatz and allow for particle-dependent coefficients; hence allowing different magnitudes of Lorentz symmetry violation for different particles even when considering the same order terms (same n) in regards to momentum. Any violation of Lorentz invariance should be due to the microscopic structure of the effective space-time. This implies that one has to tune the system in order to cancel exactly all those violations of Lorentz invariance which are solely due to mode-mixing interactions in the hydrodynamic limit.

Super-Poincaré Algebra: Is There a Case for Nontrivial Geometry in the Bosonic Sector?

In one dimension there is no Lorentz group and therefore all bosonic and fermionic fields have no space-time indices. The simplest free action for one bosonic field φ and one fermionic field ψ reads

S = γ∫dt [φ^.2 – i/2ψψ^.] —– (1)

We treat the scalar field as dimensionless and assign dimension cm^−1/2 to fermions. Therefore, all our actions will contain the parameter γ with the dimension [γ] = cm. (1) provides the first example of a supersymmetric invariant action and is invariant w.r.t. the following transformations:

δφ = −iεψ, δψ = −εφ ̇ —– (2)

The infinitesimal parameter ε anticommutes with fermionic fields and with itself. What is really important about transformations (2) is their commutator

δ₂δ₁φ = δ₂(−ε₁ψ) = iε₁ε₂φ ̇

δ₁δ₂φ = iε₂ε₁φ ̇ ⇒ [δ₂, δ₁] φ = 2iε₁ε₂φ ̇ —– (3)

Thus, from (3) we may see the main property of supersymmetry transformations: they commute on translations. In our simplest one-dimensional framework this is the time translation. This property has the followin form in terms of the supersymmetry generator Q:

{Q, Q} = −2P —– (4)

The anticommutator (4), together with

[Q, P] = 0 —– (5)

describe N = 1 super-Poincaré algebra in d = 1. The structure of N-extended super-Poincaré algebra includes N real super-charges Q^A , A = 1, . . . , N with the following anti commutators:

{Q^A, Q^B} = −2δ^ABP, [Q^A, P] = 0 —– (6)

Let us stress that the reality of the supercharges is very important, as well as having the same sign in the r.h.s. of Q^A, Q^B ∀ Q^A.

From (2) we see that the minimal N = 1 supermultiplet includes one bosonic and one fermionic field. A natural question arises: how many components do we need, in order to realize the N-extended superalgebra (6)? In order to mimic the transformations (2) for all N supertranslations

δφ_i = −iε^A(L_A)^iˆ_i ψ_iˆ, δψ_iˆ = −ε^A (R_A)^iˆ_iφ ̇_i —– (7)

Here the indices i = 1,…,d_b and iˆ = 1,…,d_f count the numbers of bosonic and fermionic components, while (L_A)^iˆ_i and (R_A)ⁱ_iˆ are N arbitrary, for the time being, matrices. The additional conditions one should impose on the transformations (7) are

• they should form the N-extended superalgebra (6)
• they should leave invariant the free action constructed from the involved fields.

When N > 8 the minimal dimension of the supermultiplets rapidly increases and the analysis of the corresponding theories becomes very complicated. For many reasons, the most interesting case seems to be the N = 8 supersymmetric mechanics. Being the highest N case of minimal N-extended supersymmetric mechanics admitting realization on N bosons (physical and auxiliary) and N fermions, the systems with eight supercharges are the highest N ones, among the extended supersymmetric systems, which still possess a nontrivial geometry in the bosonic sector. When the number of supercharges exceeds 8, the target spaces are restricted to be symmetric spaces.

Philosophizing Loops – Why Spin Foam Constraints to 3D Dynamics Evolution?

The philosophy of loops is canonical, i.e., an analysis of the evolution of variables defined classically through a foliation of spacetime by a family of space-like three- surfaces ∑_t. The standard choice is the three-dimensional metric g_ij, and its canonical conjugate, related to the extrinsic curvature. If the system is reparametrization invariant, the total hamiltonian vanishes, and this hamiltonian constraint is usually called the Wheeler-DeWitt equation. Choosing the canonical variables is fundamental, to say the least.

Abhay Ashtekar‘s insights stems from the definition of an original set of variables stemming from Einstein-Hilbert Lagrangian written in the form,

S = ∫e^a ∧ e^b ∧ R^cdε_abcd —– (1)

where, e^a are the one-forms associated to the tetrad,

e^a ≡ e^a_μdx^μ —– (2)

The associated SO(1, 3) connection one-form ϖ^a_b is called the spin connection. Its field strength is the curvature expressed as a two form:

R^a_b ≡ dϖ^a_b + ϖ^a_c ∧ ϖ^c_b —– (3)

Ashtekar’s variables are actually based on the SU(2) self-dual connection

A = ϖ − i ∗ ϖ —– (4)

Its field strength is

F ≡ dA + A ∧ A —– (5)

The dynamical variables are then (A_i, Eⁱ ≡ F⁰ⁱ). The main virtue of these variables is that constraints are then linearized. One of them is exactly analogous to Gauss’ law:

D_iEⁱ = 0 —– (6)

There is another one related to three-dimensional diffeomorphisms invariance,

trF_ijEⁱ = 0 —– (7)

and, finally, there is the Hamiltonian constraint,

trF_ijEⁱE^j = 0 —– (8)

On a purely mathematical basis, there is no doubt that Astekhar’s variables are of a great ingenuity. As a physical tool to describe the metric of space, they are not real in general. This forces a reality condition to be imposed, which is akward. For this reason it is usually prefered to use the Barbero-Immirzi formalism in which the connection depends on a free parameter, γ

Aⁱ_a + ϖⁱ_a + γKⁱ_a —– (9)

ϖ being the spin connection, and K the extrinsic curvature. When γ = i, Ashtekar’s formalism is recovered, for other values of γ, the explicit form of the constraints is more complicated. Even if there is a Hamiltonian constraint that seems promising, was isn’t particularly clear is if the quantum constraint algebra is isomorphic to the classical algebra.

Some states which satisfy the Astekhar constraints are given by the loop representation, which can be introduced from the construct (depending both on the gauge field A and on a parametrized loop γ)

W (γ, A) ≡ trPe^φ_γA —– (10)

and a functional transform mapping functionals of the gauge field ψ(A) into functionals of loops, ψ(γ):

ψ(γ) ≡ ∫DAW(γ, A) ψ(A) —– (11)

When one divides by diffeomorphisms, it is found that functions of knot classes (diffeomorphisms classes of smooth, non self-intersecting loops) satisfy all the constraints. Some particular states sought to reproduce smooth spaces at coarse graining are the Weaves. It is not clear to what extent they also approach the conjugate variables (that is, the extrinsic curvature) as well.

In the presence of a cosmological constant the hamiltonian constraint reads:

ε_ijkE^aiE^bj(F^k_ab + λ/3ε_abcE^ck) = 0 —– (12)

A particular class of solutions expounded by Lee Smolin of the constraint are self-dual solutions of the form

Fⁱ_ab = -λ/3ε_abcE^ci —– (13)

Loop states in general (suitable symmetrized) can be represented as spin network states: colored lines (carrying some SU(2) representation) meeting at nodes where intertwining SU(2) operators act. There is also a path integral representation, known as spin foam, a topological theory of colored surfaces representing the evolution of a spin network. Spin foams can also be considered as an independent approach to the quantization of the gravitational field. In addition to its specific problems, the hamiltonian constraint does not say in what sense (with respect to what) the three-dimensional dynamics evolve.

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.

Killing Fields

Let κ^a be a smooth field on our background spacetime (M, g_ab). κ^a is said to be a Killing field if its associated local flow maps Γ_s are all isometries or, equivalently, if £_κ g_ab = 0. The latter condition can also be expressed as ∇(_aκ_b) = 0.

Any number of standard symmetry conditions—local versions of them, at least can be cast as claims about the existence of Killing fields. Local, because killing fields need not be complete, and their associated flow maps need not be defined globally.

(M, g_ab) is stationary if it has a Killing field that is everywhere timelike.

(M, g_ab) is static if it has a Killing field that is everywhere timelike and locally hypersurface orthogonal.

(M, g_ab) is homogeneous if its Killing fields, at every point of M, span the tangent space.

In a stationary spacetime there is, at least locally, a “timelike flow” that preserves all spacetime distances. But the flow can exhibit rotation. Think of a whirlpool. It is the latter possibility that is ruled out when one passes to a static spacetime. For example, Gödel spacetime, is stationary but not static.

Let κ^a be a Killing field in an arbitrary spacetime (M, g_ab) (not necessarily Minkowski spacetime), and let γ : I → M be a smooth, future-directed, timelike curve, with unit tangent field ξ^a. We take its image to represent the worldline of a point particle with mass m > 0. Consider the quantity J = (P^aκ_a), where P^a = mξ^a is the four-momentum of the particle. It certainly need not be constant on γ[I]. But it will be if γ is a geodesic. For in that case, ξⁿ∇_nξ^a = 0 and hence

ξⁿ∇_nJ = m(κ_a ξⁿ∇_nξ^a + ξⁿξ^a ∇_nκ_a) = mξⁿξ^a ∇(_nκ_a) = 0

Thus, J is constant along the worldlines of free particles of positive mass. We refer to J as the conserved quantity associated with κ^a. If κ^a is timelike, we call J the energy of the particle (associated with κ^a). If it is spacelike, and if its associated flow maps resemble translations, we call J the linear momentum of the particle (associated with κ^a). Finally, if κ^a is spacelike, and if its associated flow maps resemble rotations, then we call J the angular momentum of the particle (associated with κ^a).

It is useful to keep in mind a certain picture that helps one “see” why the angular momentum of free particles (to take that example) is conserved. It involves an analogue of angular momentum in Euclidean plane geometry. Figure below shows a rotational Killing field κ^a in the Euclidean plane, the image of a geodesic (i.e., a line) L, and the tangent field ξ^a to the geodesic. Consider the quantity J = ξ^aκ_a, i.e., the inner product of ξ^a with κ^a – along L, and we can better visualize the assertion.

Figure: κ^a is a rotational Killing field. (It is everywhere orthogonal to a circle radius, and is proportional to it in length.) ξ^a is a tangent vector field of constant length on the line L. The inner product between them is constant. (Equivalently, the length of the projection of κ^a onto the line is constant.)

Let us temporarily drop indices and write κ·ξ as one would in ordinary Euclidean vector calculus (rather than ξ^aκ_a). Let p be the point on L that is closest to the center point where κ vanishes. At that point, κ is parallel to ξ. As one moves away from p along L, in either direction, the length ∥κ∥ of κ grows, but the angle ∠(κ,ξ) between the vectors increases as well. It should seem at least plausible from the picture that the length of the projection of κ onto the line is constant and, hence, that the inner product κ·ξ = cos(∠(κ , ξ )) ∥κ ∥ ∥ξ ∥ is constant.

That is how to think about the conservation of angular momentum for free particles in relativity theory. It does not matter that in the latter context we are dealing with a Lorentzian metric and allowing for curvature. The claim is still that a certain inner product of vector fields remains constant along a geodesic, and we can still think of that constancy as arising from a compensatory balance of two factors.

Let us now turn to the second type of conserved quantity, the one that is an attribute of extended bodies. Let κ^a be an arbitrary Killing field, and let T_ab be the energy-momentum field associated with some matter field. Assume it satisfies the conservation condition (∇_aT^ab = 0). Then (T^abκ_b) is divergence free:

∇_a(T^abκ_b) = κ_b∇_aT^ab + T^ab∇_aκ_b = T^ab∇(_aκ_b) = 0

(The second equality follows from the conservation condition and the symmetry of T^ab; the third follows from the fact that κ^a is a Killing field.) It is natural, then, to apply Stokes’s theorem to the vector field (T^abκ_b). Consider a bounded system with aggregate energy-momentum field T_ab in an otherwise empty universe. Then there exists a (possibly huge) timelike world tube such that T_ab vanishes outside the tube (and vanishes on its boundary).

Let S₁ and S₂ be (non-intersecting) spacelike hypersurfaces that cut the tube as in the figure below, and let N be the segment of the tube falling between them (with boundaries included).

Figure: The integrated energy (relative to a background timelike Killing field) over the intersection of the world tube with a spacelike hypersurface is independent of the choice of hypersurface.

By Stokes’s theorem,

∫_S2(T^abκ_b)dS_a – ∫_S1(T^abκ_b)dS_a = ∫_S2∩∂N(T^abκ_b)dS_a – ∫_S1∩∂N(T^abκ_b)dS_a

= ∫_∂N(T^abκ_b)dS_a = ∫_N∇_a(T^abκ_b)dV = 0

Thus, the integral ∫_S(T^abκ_b)dS_a is independent of the choice of spacelike hypersurface S intersecting the world tube, and is, in this sense, a conserved quantity (construed as an attribute of the system confined to the tube). An “early” intersection yields the same value as a “late” one. Again, the character of the background Killing field κ^a determines our description of the conserved quantity in question. If κ^a is timelike, we take ∫_S(T^abκ_b)dS_a to be the aggregate energy of the system (associated with κ^a). And so forth.

Geometry and Localization: An Unholy Alliance? Thought of the Day 95.0

There are many misleading metaphors obtained from naively identifying geometry with localization. One which is very close to that of String Theory is the idea that one can embed a lower dimensional Quantum Field Theory (QFT) into a higher dimensional one. This is not possible, but what one can do is restrict a QFT on a spacetime manifold to a submanifold. However if the submanifold contains the time axis (a ”brane”), the restricted theory has too many degrees of freedom in order to merit the name ”physical”, namely it contains as many as the unrestricted; the naive idea that by using a subspace one only gets a fraction of phase space degrees of freedom is a delusion, this can only happen if the subspace does not contain a timelike line as for a null-surface (holographic projection onto a horizon).

The geometric picture of a string in terms of a multi-component conformal field theory is that of an embedding of an n-component chiral theory into its n-dimensional component space (referred to as a target space), which is certainly a string. But this is not what modular localization reveals, rather those oscillatory degrees of freedom of the multicomponent chiral current go into an infinite dimensional Hilbert space over one localization point and do not arrange themselves according according to the geometric source-target idea. A theory of this kind is of course consistent but String Theory is certainly a very misleading terminology for this state of affairs. Any attempt to imitate Feynman rules by replacing word lines by word sheets (of strings) may produce prescriptions for cooking up some mathematically interesting functions, but those results can not be brought into the only form which counts in a quantum theory, namely a perturbative approach in terms of operators and states.

String Theory is by no means the only area in particle theory where geometry and modular localization are at loggerheads. Closely related is the interpretation of the Riemann surfaces, which result from the analytic continuation of chiral theories on the lightray/circle, as the ”living space” in the sense of localization. The mathematical theory of Riemann surfaces does not specify how it should be realized; if its refers to surfaces in an ambient space, a distinguished subgroup of Fuchsian group or any other of the many possible realizations is of no concern for a mathematician. But in the context of chiral models it is important not to confuse the living space of a QFT with its analytic continuation.

Whereas geometry as a mathematical discipline does not care about how it is concretely realized the geometrical aspects of modular localization in spacetime has a very specific geometric content namely that which can be encoded in subspaces (Reeh-Schlieder spaces) generated by operator subalgebras acting onto the vacuum reference state. In other words the physically relevant spacetime geometry and the symmetry group of the vacuum is contained in the abstract positioning of certain subalgebras in a common Hilbert space and not that which comes with classical theories.