Schematic Grothendieck Representation

A spectral Grothendieck representation Rep is said to be schematic if for every triple γ ≤ τ ≤ δ in Top(A), for every A in R^(Ring) we have a commutative diagram in R^:

IMG_20191226_064217

 

If Rep is schematic, then, P : Top(A) → R^ is a presheaf with values in R^ over the lattice Top(A)o, for every A in R.

The modality is to restrict attention to Tors(Rep(A)); that is, a lattice in the usual sense; and hence this should be viewed as the commutative shadow of a suitable noncommutative theory.

For obtaining the complete lattice Q(A), a duality is expressed by an order-reversing bijection: (−)−1 : Q(A) → Q((Rep(A))o). (Rep(A))o is not a Grothendieck category. It is additive and has a projective generator; moreover, it is known to be a varietal category (also called triplable) in the sense that it has a projective regular generator P, it is co-complete and has kernel pairs with respect to the functor Hom(P, −), and moreover every equivalence relation in the category is a kernel pair. If a comparison functor is constructed via Hom(P, −) as a functor to the category of sets, it works well for the category of set-valued sheaves over a Grothendieck topology.

Now (−)−1 is defined as an order-reversing bijection between idempotent radicals on Rep(A) and (Rep(A))o, implying we write (Top(A))−1 for the image of Top(A) in Q((Rep( A))o). This is encoded in the exact sequence in Rep(A):

0 → ρ(M) → M → ρ−1(M) → 0

(reversed in (Rep(A))o). By restricting attention to hereditary torsion theories (kernel functors) when defining Tors(−), we introduce an asymmetry that breaks the duality because Top(A)−1 is not in Tors((Rep(A))op). If notationally, TT(G) is the complete lattice of torsion theories (not necessarily hereditary) of the category G; then (TT(G))−1 ≅ TT(Gop). Hence we may view Tors(G)−1 as a complete sublattice of TT(Gop).

Black Hole Entropy in terms of Mass. Note Quote.

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If M-theory is compactified on a d-torus it becomes a D = 11 – d dimensional theory with Newton constant

GD = G11/Ld = l911/Ld —– (1)

A Schwartzschild black hole of mass M has a radius

Rs ~ M(1/(D-3)) GD(1/(D-3)) —– (2)

According to Bekenstein and Hawking the entropy of such a black hole is

S = Area/4GD —– (3)

where Area refers to the D – 2 dimensional hypervolume of the horizon:

Area ~ RsD-2 —– (4)

Thus

S ~ 1/GD (MGD)(D-2)/(D-3) ~ M(D-2)/(D-3) GD1/(D-3) —– (5)

From the traditional relativists’ point of view, black holes are extremely mysterious objects. They are described by unique classical solutions of Einstein’s equations. All perturbations quickly die away leaving a featureless “bald” black hole with ”no hair”. On the other hand Bekenstein and Hawking have given persuasive arguments that black holes possess thermodynamic entropy and temperature which point to the existence of a hidden microstructure. In particular, entropy generally represents the counting of hidden microstates which are invisible in a coarse grained description. An ultimate exact treatment of objects in matrix theory requires a passage to the infinite N limit. Unfortunately this limit is extremely difficult. For the study of Schwarzchild black holes, the optimal value of N (the value which is large enough to obtain an adequate description without involving many redundant variables) is of order the entropy, S, of the black hole.

Considering the minimum such value for N, we have

Nmin(S) = MRs = M(MGD)1/D-3 = S —– (6)

We see that the value of Nmin in every dimension is proportional to the entropy of the black hole. The thermodynamic properties of super Yang Mills theory can be estimated by standard arguments only if S ≤ N. Thus we are caught between conflicting requirements. For N >> S we don’t have tools to compute. For N ~ S the black hole will not fit into the compact geometry. Therefore we are forced to study the black hole using N = Nmin = S.

Matrix theory compactified on a d-torus is described by d + 1 super Yang Mills theory with 16 real supercharges. For d = 3 we are dealing with a very well known and special quantum field theory. In the standard 3+1 dimensional terminology it is U(N) Yang Mills theory with 4 supersymmetries and with all fields in the adjoint repersentation. This theory is very special in that, in addition to having electric/magnetic duality, it enjoys another property which makes it especially easy to analyze, namely it is exactly scale invariant.

Let us begin by considering it in the thermodynamic limit. The theory is characterized by a “moduli” space defined by the expectation values of the scalar fields φ. Since the φ also represents the positions of the original DO-branes in the non compact directions, we choose them at the origin. This represents the fact that we are considering a single compact object – the black hole- and not several disconnected pieces.

The equation of state of the system, defined by giving the entropy S as a function of temperature. Since entropy is extensive, it is proportional to the volume ∑3 of the dual torus. Furthermore, the scale invariance insures that S has the form

S = constant T33 —– (7)

The constant in this equation counts the number of degrees of freedom. For vanishing coupling constant, the theory is described by free quanta in the adjoint of U(N). This means that the number of degrees of freedom is ~ N2.

From the standard thermodynamic relation,

dE = TdS —– (8)

and the energy of the system is

E ~ N2T43 —– (9)

In order to relate entropy and mass of the black hole, let us eliminate temperature from (7) and (9).

S = N23((E/N23))3/4 —– (10)

Now the energy of the quantum field theory is identified with the light cone energy of the system of DO-branes forming the black hole. That is

E ≈ M2/N R —– (11)

Plugging (11) into (10)

S = N23(M2R/N23)3/4 —– (12)

This makes sense only when N << S, as when N >> S computing the equation of state is slightly trickier. At N ~ S, this is precisely the correct form for the black hole entropy in terms of the mass.

Catastrophe, Gestalt and Thom’s Natural Philosophy of 3-D Space as Underlying All Abstract Forms – Thought of the Day 157.0

The main result of mathematical catastrophe theory consists in the classification of unfoldings (= evolutions around the center (the germ) of a dynamic system after its destabilization). The classification depends on two sorts of variables:

(a) The set of internal variables (= variables already contained in the germ of the dynamic system). The cardinal of this set is called corank,

(b) the set of external variables (= variables governing the evolution of the system). Its cardinal is called codimension.

The table below shows the elementary catastrophes for Thom:

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The A-unfoldings are called cuspoids, the D-unfoldings umbilics. Applications of the E-unfoldings have only been considered in A geometric model of anorexia and its treatment. By loosening the condition for topological equivalence of unfoldings, we can enlarge the list, taking in the family of double cusps (X9) which has codimension 8. The unfoldings A3(the cusp) and A5 (the butterfly) have a positive and a negative variant A+3, A-3, A+5, A-5.

We obtain Thorn’s original list of seven “catastrophes” if we consider only unfoldings up to codimension 4 and without the negative variants of A3 and A5.

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Thom argues that “gestalts” are locally con­stituted by maximally four disjoint constituents which have a common point of equilibrium, a common origin. This restriction is ultimately founded in Gibb’s law of phases, which states that in three-dimensional space maximally four independent systems can be in equilibrium. In Thom’s natural philosophy, three-dimensional space is underlying all abstract forms. He, therefore, presumes that the restriction to four constituents in a “gestalt” is a kind of cognitive universal. In spite of the plausibility of Thom’s arguments there is a weaker assumption that the number of constituents in a gestalt should be finite and small. All unfoldings with codimension (i.e. number of external variables) smaller than or equal to 5 have simple germs. The unfoldings with corank (i.e. number of internal variables) greater than two have moduli. As a matter of fact the most prominent semantic archetypes will come from those unfoldings considered by René Thom in his sketch of catastrophe theoretic semantics.

Man Proposes OR Woman Proposes – “Matches” Happen Algorithmically and are Economically Walrasian Rather than Keynesian. Note Quote & Didactics.

Consider a set M of men and a set W of women. Each m ∈ M has a strict preference ordering over the elements of W and each w ∈ W has a strict preference ordering over men. Let us denote the preference ordering of an agent i by i and x ≻i y will mean that agent i ranks x above y. Now a marriage or matching would be considered as an assignment of men to women such that each man is assigned to at most one woman and vice versa. But, what if the agent decides to remain single. This is possible by two ways, viz. if a man or a woman is matched with oneself, or for each man or woman, there is a dummy woman or man in the set W or M that corresponds to being single. If this were the construction, then, we could safely assume |M| = |W|. But, there is another impediment here, whereby a subversion of sorts is possible, in that a group of agents could simply opt out of the matching exercise. In such a scenario, it becomes mandatory to define a blocking set. As an implication of such subversiveness, a matching is called unstable if there are two men m, m’ and two women w, w’ such that

  1. m is matched to w
  2. m’ is matched to w’, and
  3. w’ m w and m ≻w’ m’

then, the pair (m, w’) is a blocking pair. Any matching without the blocking pair is called stable.

Now, given the preferences of men and women, is it always possible to find stable matchings? For the same, what is used is Gale and Shapley’s deferred acceptance algorithm.

So, after a brief detour, let us concentrate on the male-proposal version.

First, each man proposes to his top-ranked choice. Next, each woman who has received at least two proposals keeps (tentatively) her top-ranked proposal and rejects the rest. Then, each man who has been rejected proposes to his top-ranked choice among the women who have not rejected him. Again each woman who has at least two proposals (including ones from previous rounds) keeps her top-ranked proposal and rejects the rest. The process repeats until no man has a woman to propose to or each woman has at most one proposal. At this point the algorithm terminates and each man is assigned to a woman who has not rejected his proposal. No man is assigned to more than one woman. Since each woman is allowed to keep only one proposal at any stage, no woman is assigned to more than one man. Therefore the algorithm terminates in a matching.

IMG_20190909_141003

Consider the matching {(m1, w1), (m2, w2), (m3, w3)}. This is an unstable matching since (m1, w2) is a blocking pair. The matching {(m1, w1), (m3, w2), (m2, w3)}, however, is stable. Now looking at the figure above, m1 proposes to w2, m2 to w1, and m3 to w1. At the end of this round, w1 is the only woman to have received two proposals. One from m3 and the other from m2. Since she ranks m3 above m2, she keeps m3 and rejects m2. Since m3 is the only man to have been rejected, he is the only one to propose again in the second round. This time he proposes to w3. Now each woman has only one proposal and the algorithm terminates with the matching {(m1, w2), (m2, w3), (m3, w2)}.

The male propose algorithm terminates in a stable matching.

Suppose not. Then ∃ a blocking pair (m1, w1) with m1 matched to w2, say, and w1 matched to m2. Since (m1, w1) is blocking and w1m1 w2, in the proposal algorithm, m1 would have proposed to w1 before w2. Since m1 was not matched with w1 by the algorithm, it must be because w1 received a proposal from a man that she ranked higher than m1. Since the algorithm matches her to m2 it follows that m2w1 m1. This contradicts the fact that (m1, w1) is a blocking pair.

Even if where the women propose, the outcome would still be stable matching. The only difference is in kind as the stable matching would be different from the one generated when the men propose. This would also imply that even if stable matching is guaranteed to exist, there is more than one such matching. Then what is the point to prefer one to the other? Well, there is a reason:

Denote a matching by μ. The woman assigned to man m in the matching μ is denoted μ(m). Similarly, μ(w) is the man assigned to woman w. A matching μ is male-optimal if there is no stable matching ν such that ν(m) ≻m μ(m) or ν(m) = μ(m) ∀ m with ν(j) ≻j μ(j) for at least one j ∈ M. Similarly for the female-optimality.

The stable matching produced by the (male-proposal) Deferred Acceptance Algorithm is male-optimal.

Let μ be the matching returned by the male-propose algorithm. Suppose μ is not male optimal. Then, there is a stable matching ν such that ν(m) ≻m μ(m) or ν(m) = μ(m) ∀ m with ν(j) ≻j μ(j) for at least one j ∈ M. Therefore, in the application of the proposal algorithm, there must be an iteration where some man j proposes to ν(j) before μ(j) since ν(j) ≻j μ(j) and is rejected by woman ν(j). Consider the first such iteration. Since woman ν(j) rejects j she must have received a proposal from a man i she prefers to man j. Since this is the first iteration at which a male is rejected by his partner under ν, it follows that man i ranks woman ν(j) higher than ν(i). Summarizing, i ≻ν(j) j and ν(j) ≻i ν(i) implying that ν is not stable, a contradiction.

Now, the obvious question is if this stable matching is optimal w.r.t. to both men and women? The answer this time around is NO. From above, it could easily be seen that there are two stable matchings, one of them is male-optimal and the other is female-optimal. At least, one female is strictly better-off under the female optimality than male optimality, and by this, no female is worse off. If the POV is men, a similar conclusion is drawn.  A stable marriage is immune to a pair of agents opting out of the matching. We could ask that no subset of agents should have an incentive to opt out of the matching. Formally, a matching μ′ dominates a matching μ if there is a set S ⊂ M ∪ W such that for all m, w ∈ S, both (i) μ′(m), μ′(w) ∈ S and (ii) μ′(m) ≻m μ(m) and μ′(w) ≻w μ(w). Stability is a special case of this dominance condition when we restrict attention to sets S consisting of a single couple. The set of undominated matchings is called the core of the matching game.

The direct mechanism associated with the male propose algorithm is strategy-proof for the males.

Suppose not. Then there is a profile of preferences π = (≻m1 , ≻m2 , . . . , ≻mn) for the men, such that man m1, say, can misreport his preferences and obtain a better match. To express this formally, let μ be the stable matching obtained by applying the male proposal algorithm to the profile π. Suppose that m1 reports the preference ordering ≻ instead. Let ν be the stable matching that results when the male-proposal algorithm is applied to the profile π1 = (≻, ≻m2 , . . . , ≻mn). For a contradiction, suppose ν(m1) ≻m1 μ(m1). For notational convenience let a ≽m b mean that a ≻m b or a = b.

First we show that m1 can achieve the same effect by choosing an ordering ≻̄ where woman ν(m1) is ranked first. Let π2 = (≻̄ , ≻m2 , . . . , ≻mn). Knowing that ν is stable w.r.t the profile π1 we show that it is stable with respect to the profile π2. Suppose not. Then under the profile π2 there must be a pair (m, w) that blocks ν. Since ν assigns to m1 its top choice with respect to π2, m1 cannot be part of this blocking pair. Now the preferences of all agents other than m1 are the same in π1 and π2. Therefore, if (m, w) blocks ν w.r.t the profile π2, it must block ν w.r.t the profile π1, contradicting the fact that ν is a stable matching under π1.

Let λ be the male propose stable matching for the profile π2. ν is a stable matching w.r.t the profile π2. As λ is male optimal w.r.t the profile π2, it follows that λ(m1) = ν(m1).

Let’s assume that ν(m1) is the top-ranked woman in the ordering ≻. Now we show that the set B = {mj : μ(mj) ≻mj ν(mj)} is empty. This means that all men, not just m1, are no worse off under ν compared to μ. Since ν is stable w.r.t the original profile, π this contradicts the male optimality of μ.

Suppose B ≠ ∅. Therefore, when the male proposal algorithm is applied to the profile π1, each mj ∈ B is rejected by their match under μ, i.e., μ(mj). Consider the first iteration of the proposal algorithm where some mj is rejected by μ(mj). This means that woman μ(mj) has a proposal from man mk that she ranks higher, i.e., mkμ(mj) mj. Since mk was not matched to μ(mj) under μ it must be that μ(mk) ≻mk μ(mj). Hence mk ∈ B , otherwise μ(mj) ≽ mkν(mk) ≽mk μ(mk) ≻mk μ(mj), which is a contradiction. Since mk ∈ B and mk has proposed to μ(mj) at the time man mj proposes, it means that mk must have been rejected by μ(mk) prior to mj being rejected, contradicting our choice of mj.

The mechanism associated with the male propose algorithm is not strategy-proof for the females. Let us see how this is the case by way of an example. The male propose algorithm returns the matching {(m1, w2), (m2, w3), (m3, w1)}. In the course of the algorithm the only woman who receives at least two proposals is w1. She received proposals from m2 and m3. She rejects m2 who goes on to propose to w3 and the algorithm terminates. Notice that w1 is matched with her second choice. Suppose now that she had rejected m3 instead. Then m3 would have gone on to proposes to w2. Woman w2 now has a choice between m1 and m3. She would keep m3 and reject m1, who would go on to propose to w1. Woman w1 would keep m1 over m2 and in the final matching be paired with a her first-rank choice.

Transposing this on to economic theory, this fits neatly into the Walrasian equilibrium. Walras’ law is an economic theory that the existence of excess supply in one market must be matched by excess demand in another market so that it balances out. Walras’ law asserts that an examined market must be in equilibrium if all other markets are in equilibrium, and this contrasts with Keynesianism, which by contrast, assumes that it is possible for just one market to be out of balance without a “matching” imbalance elsewhere. Moreover, Walrasian equilibria are the solutions of a fixed point problem. In the cases when they can be computed efficiently it is because the set of Walrasian equilibria can be described by a set of convex inequalities. The same can be said of stable matchings/marriages. The set of stable matchings is fixed points of a nondecreasing function defined on a lattice. 

AI learns to solve a Rubik’s Cube in 1.2 seconds

DeepCubeA uses a neural network (which apes how the human mind processes information) along with machine learning techniques, in which an AI system learns by detecting patterns and theorizing with little human input. It adopts a reinforcement learning approach, by which it learned “how to solve increasingly difficult states in reverse from the goal state without any specific domain knowledge.”

Endgadget

ABSTRACT

 

Recently, Approximate Policy Iteration (API) algorithms have achieved super- human proficiency in two-player zero-sum games such as Go, Chess, and Shogi without human data. These API algorithms iterate between two policies: a slow policy (tree search), and a fast policy (a neural network). In these two-player games, a reward is always received at the end of the game. However, the Rubik’s Cube has only a single solved state, and episodes are not guaranteed to terminate. This poses a major problem for these API algorithms since they rely on the reward received at the end of the game. We introduce Autodidactic Iteration: an API algorithm that overcomes the problem of sparse rewards by training on a distribution of states that allows the reward to propagate from the goal state to states farther away. Autodi- dactic Iteration is able to learn how to solve the Rubik’s Cube without relying on human data. Our algorithm is able to solve 100% of randomly scrambled cubes while achieving a median solve length of 30 moves — less than or equal to solvers that employ human domain knowledge.

 

SOLVING THE RUBIK’S CUBE WITH APPROXIMATE POLICY ITERATION

Homological Algebra – Does A∞ Algebra Compensate for any Loss of Information in the Study of Chain Complexes? 1.0

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In an abelian category, homological algebra is the homotopy theory of chain complexes up to quasi-isomorphism of chain complexes.  When considering nonnegatively graded chain complexes, homological algebra may be viewed as a linearized version of the homotopy theory of homotopy types or infinite groupoids. When considering unbounded chain complexes, it may be viewed as a linearized and stabilized version. Conversely, we may view homotopical algebra as a nonabelian generalization of homological algebra.

Suppose we have a topological space X and a “multiplication map” m2 : X × X → X. This map may or may not be associative; imposing associativity is an extra condition. An A space imposes a weaker structure, which requires m2 to be associative up to homotopy, along with “higher order” versions of this. Indeed, there are very standard situations where one has natural multiplication maps which are not associative, but obey certain weaker conditions.

The standard example is when X is the loop space of another space M, i.e., if m0 ∈ M is a chosen base point,

X = {x : [0,1] → M |x continuous, x(0) = x(1) = m0}.

Composition of loops is then defined, with

x2x1(t) = x2(2t), when 0 ≤ t ≤ 1/2

= x1(2t−1), when  1/2 ≤ t ≤ 1

However, this composition is not associative, but x3(x2x1) and (x1x2)x3 are homotopic loops.

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On the left, we first traverse x3 from time 0 to time 1/2, then traverse x2 from time 1/2 to time 3/4, and then x1 from time 3/4 to time 1. On the right, we first traverse x3 from time 0 to time 1/4, x2 from time 1/4 to time 1/2, and then x1 from time 1/2 to time 1. By continuously deforming these times, we can homotop one of the loops to the other. This homotopy can be represented by a map

m3 : [0, 1] × X × X × X → X such that

{0} × X × X × X → X is given by (x3, x2, x1) 􏰀→ m2(x3, m2(x2, x1)) and

{1} × X × X × X → X is given by (x3, x2, x1) 􏰀→ m2(m2(x3, x2), x1)

What, if we have four elements x1, . . . , x4 of X? Then there are a number of different ways of putting brackets in their product, and these are related by the homotopies defined by m3. Indeed, we can relate

((x4x3)x2)x1 and x4(x3(x2x1))

in two different ways:

((x4x3)x2)x1 ∼ (x4x3)(x2x1) ∼ x4(x3(x2x1))

and

((x4x3)x2)x1 ∼ (x4(x3x2))x1 ∼ x4((x3x2)x1) ∼ x4(x3(x2x1)).

Here each ∼ represents a homotopy given by m3.

Schematically, this is represented by a polygon, S4, with each vertex labelled by one of the ways of associating x4x3x2x1, and the edges represent homotopies between them

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The homotopies myield a map ∂S4 × X4 → X which is defined using appropriate combinations of m2 and m3 on each edge of the boundary of S4. For example, restricting to the edge with vertices ((x4x3)x2)x1 and (x4(x3x2))x1, this map is given by (s, x4, . . . , x1) 􏰀→ m2(m3(s, x4, x3, x2), x1).

Thus the conditionality on the structure becomes: this map extend across S4, giving a map

m4 : S4 × X4 → X.

As homological algebra seeks to study complexes by taking quotient modules to obtain the homology, the question arises as to whether any information is lost in this process. This is equivalent to asking whether it is possible to reconstruct the original complex (up to quasi-isomorphism) given its homology or whether some additional structure is needed in order to be able to do this. The additional structure that is needed is an A-structure constructed on the homology of the complex…

 

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

Particle-b-interacts-with-particle-a-at-point-x-via-retarded-and-advanced-waves-The-mass

In other words, if zi􏱁i)  and zjj􏱁) 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

(zμi zμj)(zμi zμj) = (zi – zj)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 Fμν, is described by Maxwell’s equations. The homogenous Maxwell equations tell us that the antisymmetric tensor Fμν (a 2-form) can be written as the exterior derivative of a potential (a 1-form) Aμ(x), i.e. as

Fμν = ∂μ Aν – ∂ν Aμ —– (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 ∂μAμ(x) = 0 (Lorentz gauge), the remaining Maxwell equations take the particularly simple form:

□ Aμ = – 4π jμ —– (3)

where

□ = ∂μμ

is the d’Alembert operator and jμ 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 – xi(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.