# Killing Fields

Let κa be a smooth field on our background spacetime (M, gab). κa is said to be a Killing field if its associated local flow maps Γs are all isometries or, equivalently, if £κ gab = 0. The latter condition can also be expressed as ∇(aκb) = 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, gab) is stationary if it has a Killing field that is everywhere timelike.

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

(M, gab) 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 κa be a Killing field in an arbitrary spacetime (M, gab) (not necessarily Minkowski spacetime), and let γ : I → M be a smooth, future-directed, timelike curve, with unit tangent field ξa. We take its image to represent the worldline of a point particle with mass m > 0. Consider the quantity J = (Paκa), where Pa = mξa 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, ξnnξa = 0 and hence

ξnnJ = m(κa ξnnξa + ξnξanκa) = mξnξa ∇(nκa) = 0

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

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 κa in the Euclidean plane, the image of a geodesic (i.e., a line) L, and the tangent field ξa to the geodesic. Consider the quantity J = ξaκa, i.e., the inner product of ξa with κa – along L, and we can better visualize the assertion.

Figure: κa is a rotational Killing field. (It is everywhere orthogonal to a circle radius, and is proportional to it in length.) ξa 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 κa onto the line is constant.)

Let us temporarily drop indices and write κ·ξ as one would in ordinary Euclidean vector calculus (rather than ξaκa). 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 κa be an arbitrary Killing field, and let Tab be the energy-momentum field associated with some matter field. Assume it satisfies the conservation condition (∇aTab = 0). Then (Tabκb) is divergence free:

a(Tabκb) = κbaTab + Tabaκb = Tab∇(aκb) = 0

(The second equality follows from the conservation condition and the symmetry of Tab; the third follows from the fact that κa is a Killing field.) It is natural, then, to apply Stokes’s theorem to the vector field (Tabκb). Consider a bounded system with aggregate energy-momentum field Tab in an otherwise empty universe. Then there exists a (possibly huge) timelike world tube such that Tab 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(Tabκb)dSa – ∫S1(Tabκb)dSa = ∫S2∩∂N(Tabκb)dSa – ∫S1∩∂N(Tabκb)dSa

= ∫∂N(Tabκb)dSa = ∫Na(Tabκb)dV = 0

Thus, the integral ∫S(Tabκb)dSa 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 κa determines our description of the conserved quantity in question. If κa is timelike, we take ∫S(Tabκb)dSa to be the aggregate energy of the system (associated with κa). And so forth.

# Superstrings as Grand Unifier. Thought of the Day 86.0

The first step of deriving General Relativity and particle physics from a common fundamental source may lie within the quantization of the classical string action. At a given momentum, quantized strings exist only at discrete energy levels, each level containing a finite number of string states, or particle types. There are huge energy gaps between each level, which means that the directly observable particles belong to a small subset of string vibrations. In principle, a string has harmonic frequency modes ad infinitum. However, the masses of the corresponding particles get larger, and decay to lighter particles all the quicker.

Most importantly, the ground energy state of the string contains a massless, spin-two particle. There are no higher spin particles, which is fortunate since their presence would ruin the consistency of the theory. The presence of a massless spin-two particle is undesirable if string theory has the limited goal of explaining hadronic interactions. This had been the initial intention. However, attempts at a quantum field theoretic description of gravity had shown that the force-carrier of gravity, known as the graviton, had to be a massless spin-two particle. Thus, in string theory’s comeback as a potential “theory of everything,” a curse turns into a blessing.

Once again, as with the case of supersymmetry and supergravity, we have the astonishing result that quantum considerations require the existence of gravity! From this vantage point, right from the start the quantum divergences of gravity are swept away by the extended string. Rather than being mutually exclusive, as it seems at first sight, quantum physics and gravitation have a symbiotic relationship. This reinforces the idea that quantum gravity may be a mandatory step towards the unification of all forces.

Unfortunately, the ground state energy level also includes negative-mass particles, known as tachyons. Such particles have light speed as their limiting minimum speed, thus violating causality. Tachyonic particles generally suggest an instability, or possibly even an inconsistency, in a theory. Since tachyons have negative mass, an interaction involving finite input energy could result in particles of arbitrarily high energies together with arbitrarily many tachyons. There is no limit to the number of such processes, thus preventing a perturbative understanding of the theory.

An additional problem is that the string states only include bosonic particles. However, it is known that nature certainly contains fermions, such as electrons and quarks. Since supersymmetry is the invariance of a theory under the interchange of bosons and fermions, it may come as no surprise, post priori, that this is the key to resolving the second issue. As it turns out, the bosonic sector of the theory corresponds to the spacetime coordinates of a string, from the point of view of the conformal field theory living on the string worldvolume. This means that the additional fields are fermionic, so that the particle spectrum can potentially include all observable particles. In addition, the lowest energy level of a supersymmetric string is naturally massless, which eliminates the unwanted tachyons from the theory.

The inclusion of supersymmetry has some additional bonuses. Firstly, supersymmetry enforces the cancellation of zero-point energies between the bosonic and fermionic sectors. Since gravity couples to all energy, if these zero-point energies were not canceled, as in the case of non-supersymmetric particle physics, then they would have an enormous contribution to the cosmological constant. This would disagree with the observed cosmological constant being very close to zero, on the positive side, relative to the energy scales of particle physics.

Also, the weak, strong and electromagnetic couplings of the Standard Model differ by several orders of magnitude at low energies. However, at high energies, the couplings take on almost the same value, almost but not quite. It turns out that a supersymmetric extension of the Standard Model appears to render the values of the couplings identical at approximately 1016 GeV. This may be the manifestation of the fundamental unity of forces. It would appear that the “bottom-up” approach to unification is winning. That is, gravitation arises from the quantization of strings. To put it another way, supergravity is the low-energy limit of string theory, and has General Relativity as its own low-energy limit.

# Harmonies of the Orphic Mystery: Emanation of Music

As the Buddhist sage Nagarjuna states in his Seventy Verses on Sunyata, “Being does not arise, since it exists . . .” In similar fashion it can be said that mind exists, and if we human beings manifest its qualities, then the essence and characteristics of mind must be a component of our cosmic source. David Bohm’s theory of the “implicate order” within the operations of nature suggests that observed phenomena do not operate only when they become objective to our senses. Rather, they emerge out of a subjective state or condition that contains the potentials in a latent yet really existent state that is just awaiting the necessary conditions to manifest. Thus within the explicate order of things and beings in our familiar world there is the implicate order out of which all of these emerge in their own time.

Clearly, sun and its family of planets function in accordance with natural laws. The precision of the orbital and other electromagnetic processes is awesome, drawing into one operation the functions of the smallest subparticles and the largest families of sun-stars in their galaxies, and beyond even them. These individual entities are bonded together in an evident unity that we may compare with the oceans of our planet: uncountable numbers of water molecules appear to us as a single mass of substance. In seeking the ultimate particle, the building block of the cosmos, some researchers have found themselves confronted with the mystery of what it is that holds units together in an organism — any organism!

As in music where a harmony consists of many tones bearing an inherent relationship, so must there be harmony embracing all the children of cosmos. Longing for the Harmonies: Themes and Variations from Modern Physics is a book by Frank Wilczek, an eminent physicist, and his wife Betsy Devine, an engineering scientist and freelance writer. The theme of their book is set out in their first paragraph:

From Pythagoras measuring harmonies on a lyre string to R. P. Feynman beating out salsa on his bongos, many a scientist has fallen in love with music. This love is not always rewarded with perfect mastery. Albert Einstein, an ardent amateur of the violin, provoked a more competent player to bellow at him, “Einstein, can’t you count?”

Both music and scientific research, Einstein wrote, “are nourished by the same source of longing, and they complement one another in the release they offer.” It seems to us, too, that the mysterious longing behind a scientist’s search for meaning is the same that inspires creativity in music, art, or any other enterprise of the restless human spirit. And the release they offer is to inhabit, if only for a moment, some point of union between the lonely world of subjectivity and the shared universe of external reality.

In a very lucid text, Wilczek and Devine show us that the laws of nature, and the structure of the universe and all its contributing parts, can be presented in such a way that the whole compares with a musical composition comprising themes that are fused together. One of the early chapters begins with the famous lines of the great astronomer Johannes Kepler, who in 1619 referred to the music of the spheres:

The heavenly motions are nothing but a continuous song for several voices (perceived by the intellect, not by the ear); a music which, through discordant tensions, through sincopes [sic] and cadenzas, as it were (as men employ them in imitation of those natural discords) progresses towards certain pre-designed quasi six-voiced clausuras, and thereby sets landmarks in the immeasurable flow of time. — The Harmony of the World (Harmonice mundi)

Discarding the then current superstitions and misinformed speculation, through the cloud of which Kepler had to work for his insights, Wilczek and Devine note that Kepler’s obsession with the idea of the harmony of the world is actually rooted in Pythagoras’s theory that the universe is built upon number, a concept of the Orphic mystery-religions of Greece. The idea is that “the workings of the world are governed by relations of harmony and, in particular, that music is associated with the motion of the planets — the music of the spheres” (Wilczek and Devine). Arthur Koestler, in writing of Kepler and his work, claimed that the astronomer attempted

to bare the ultimate secret of the universe in an all-embracing synthesis of geometry, music, astrology, astronomy and epistemology. The Sleepwalkers

In Longing for the Harmonies the authors refer to the “music of the spheres” as a notion that in time past was “vague, mystical, and elastic.” As the foundations of music are rhythm and harmony, they remind us that Kepler saw the planets moving around the sun “to a single cosmic rhythm.” There is some evidence that he had association with a “neo-Pythagorean” movement and that, owing to the religious-fomented opposition to unorthodox beliefs, he kept his ideas hidden under allegory and metaphor.

Shakespeare, too, phrases the thought of tonal vibrations emitted by the planets and stars as the “music of the spheres,” the notes likened to those of the “heavenly choir” of cherubim. This calls to mind that Plato’s Cratylus terms the planets theoi, from theein meaning “to run, to move.” Motion does suggest animation, or beings imbued with life, and indeed the planets are living entities so much grander than human beings that the Greeks and other peoples called them “gods.” Not the physical bodies were meant, but the essence within them, in the same way that a human being is known by the inner qualities expressed through the personality.

When classical writers spoke of planets and starry entities as “animals” they did not refer to animals such as we know on Earth, but to the fact that the celestial bodies are “animated,” embodying energies received from the sun and cosmos and transmitted with their own inherent qualities added.

Many avenues open up for our reflection upon the nature of the cosmos and ourselves, and our interrelationship, as we consider the structure of natural laws as Wilczek and Devine present them. For example, the study of particles, their interactions, their harmonizing with those laws, is illuminating intrinsically and, additionally, because of their universal application. The processes involved occur here on earth, and evidently also within the solar system and beyond, explaining certain phenomena that had been awaiting clarification.

The study of atoms here on earth and their many particles and subparticles has enabled researchers to deduce how stars are born, how and why they shine, and how they die. Now some researchers are looking at what it is, whether a process or an energy, that unites the immeasurably small with the very large cosmic bodies we now know. If nature is infinite, it must be so in a qualitative sense, not merely a quantitative.

One of the questions occupying the minds of cosmologists is whether the universal energy is running down like the mechanism of an unwinding Swiss watch, or whether there is enough mass to slow the outward thrust caused by the big bang that has been assumed to have started our cosmos going. In other words, is our universe experiencing entropy — dying as its energy is being used up — or will there be a “brake” put upon the expansion that could, conceivably, result in a return to the source of the initial explosion billions of years ago? Cosmologists have been looking for enough “dark mass” to serve as such a brake.

Among the topics treated by Wilczek and Devine in threading their way through many themes and variations in modern physics, is what is known as the mass-generating Higgs field. This is a proposition formulated by Peter Higgs, a Scottish physicist, who suggests there is an electromagnetic field that pervades the cosmos and universally provides the electron particles with mass.

The background Higgs field must have very accurately the same value throughout the universe. After all, we know — from the fact that the light from distant galaxies contains the same spectral lines we find on Earth — that electrons have the same mass throughout the universe. So if electrons are getting their mass from the Higgs field, this field had better have the same strength everywhere. What is the meaning of this all-pervasive field, which exists with no apparent source? Why is it there? (Wilczek and Devine).

What is the meaning? Why is it there? These are among the most important questions that can be asked. Though physicists may provide profound mathematical equations, they will thereby offer only more precise detail as to what is happening. We shall not receive an answer to the “What” and the “Why” without recourse to meta-physics, beyond the realm of brain-devised definitions.

The human mind is limited in its present stage of evolution. It may see the logical necessity of infinity referent to space and time; for if not infinity, what then is on the other side of the “fence” that is our outermost limit? But, being able to perceive the logical necessity of infinity, the finite mind still cannot span the limitless ranges of space, time, and substance.

If we human beings are manifold in our composition, and since we draw our very existence and sustenance from the universe at large, our conjoint nature must be drawn from the sources of life, substance, and energy, in which our and all other cosmic lives are immersed.

As the authors conclude their fascinating work:

“The worlds opened to our view are graced with wonderful symmetry and uniformity. Learning to know them, to appreciate their many harmonies, is like deepening an acquaintance with some great and meaningful piece of music — surely one of the best things life has to offer.”

# Symplectic Manifolds

The canonical example of the n-symplectic manifold is that of the frame bundle, so the question is whether this formalism can be generalized to other principal bundles, and distinguished from the quantization arising from symplectic geometry on the prototype manifold, the bundle of linear frames, a good place to motivate the formalism.

Let us start with an n-dimensional manifold M, and let π : LM → M be the space of linear frames over a base manifold M, the set of pairs (m,ek), where m ∈ M and {ek},k = 1,···,n is a linear frame at m. This gives LM dimension n(n + 1), with GL(n,R) as the structure group acting freely on the right. We define local coordinates on LM in terms of those on the manifold M – for a chart on M with coordinates {xi}, let

qi(m,ek) = xi ◦ π(m,ek) = xi(m)

πji(m,ek) = ej ∂/∂xj

where {ej} denotes the coframe dual to {ej}. These coordinates are analogous to those on the cotangent bundle, except, instead of a single momentum coordinate, we now have a momentum frame. We want to place some kind of structure on LM, which is the prototype of n-symplectic geometry that is similar to symplectic geometry of the cotangent bundle T∗M. The structure equation for symplectic geometry

df= _| X dθ

gives Hamilton’s equations for the phase space of a particle, where θ is the canonical symplectic 2-form. There is a naturally defined Rn-valued 1-form on LM, the soldering form, given by

θ(X) ≡ u−1[π∗(X)] ∀X ∈ TuLM

where the point u = (m,ek) ∈ LM gives the isomorphism u : Rn → Tπ(u)M by ξiri → ξiei, where {ri} is the standard basis of Rn. The Rn-valued 2-form dθ can be shown to be non-degenerate, that is,

X _| dθ = 0 ⇔ X = 0

where we mean that each component of X dθ is identically zero. Finally, since there is also a structure group on LM, there are also group transformation properties. Let ρ be the standard representation of GL(n, R) on Rn. Then it can be shown that the pullback of dθ under right translation by g ∈ GL (n,R) is Rg dθ = ρ(g−1) · dθ.

Thus, we have an Rn-valued generalization of symplectic geometry, which motivates the following definition.

Let P be a principal fiber bundle with structure group G over an m-dimensional manifold M . Let ρ : G → GL(n, R) be a linear representation of G. An n-symplectic structure on P is a Rn-valued 2-form ω on P that is (i) closed and non-degenerate, in the sense that

X _| ω = 0 ⇔ X = 0

for a vector field X on P, and (ii) ω is equivariant, such that under the right action of G, Rg ω = ρ(g−1) · ω. The pair (P, ω) is called an n-symplectic manifold.

Here, we have modeled n-symplectic geometry after the frame bundle by defining the general n-symplectic manifold as a principal bundle. There is no reason, however, to limit ourselves to this, since we can let P be any manifold with a group action defined on it. One example of this would be to look at the action of the conformal group on R4. Since this group is locally isomorphic to O(2, 4), which is not a subgroup of GL(4, R), then forming a O(2,4) bundle over R4 cannot be thought of as simply a reduction of the frame bundle.

# Why Shouldn’t Philosophers Worry that the Detector Criterion is too Operationalist? Scattering Theory to the Rescue. Note Quote.

It is not true that a representation (K,π) of U must be a Fock representation in order for states in the Hilbert space K to have an interpretation as particle states. Indeed, one of the central tasks of “scattering theory,” is to provide criteria – in the absence of full Fock space structure – for defining particle states. These criteria are needed in order to describe scattering experiments which cannot be described in a Fock representation, but which need particle states to describe the input and output states.

Haag and Swieca propose to pick out the n-particle states by means of localized detectors; we call this the detector criterion: A state with at least n-particles is a state that would trigger n detectors that are far separated in space. Philosophers might worry that the detector criterion is too operationalist. Indeed, some might claim that detectors themselves are made out of particles, and so defining a particle in terms of a detector would be viciously circular.

If we were trying to give an analysis of the concept of a particle, then we would need to address such worries. However, scattering theory does not end with the detector criterion. Indeed, the goal is to tie the detector criterion back to some other more intrinsic definition of particle states. The traditional intrinsic definition of particle states is in terms of Wigner’s symmetry criterion:

A state of n particles (of spins si and masses mi) is a state in the tensor product of the corresponding representations of the Poincaré group.

Thus, scattering theory – as originally conceived – needs to show that the states satisfying the detector criterion correspond to an appropriate representation of the Poincaré group. In particular, the goal is to show that there are isometries Ωin, Ωout that embed Fock space F(H) into K, and that intertwine the given representations of the Poincaré group on F(H) and K.

Based on these ideas, detailed models have been worked out for the case where there is a mass gap. Unfortunately, as of yet, there is no model in which Hin = Hout, which is a necessary condition for the theory to have an S-matrix, and to define transition probabilities between incoming and outgoing states. (Here Hin is the image of Fock space in K under the isometry Ωin, and similarly for Hout.)

Buchholz and collaborators have claimed that Wigner’s symmetry criterion is too stringent – i.e. there is a more general definition of particle states. They claim that it is only by means of this more general criterion that we can solve the “infraparticles” problem, where massive particles carry a cloud of photons.

The “measurement problem” of nonrelativistic QM shows that the standard approach to the theory is impaled on the horns of a dilemma: either

(i) one must make ad hoc adjustments to the dynamics (“collapse”) when needed to explain the results of measurements, or

(ii) measurements do not, contrary to appearances, have outcomes.

There are two main responses to the dilemma: On the one hand, some suggest that we abandon the unitary dynamics of QM in favor of stochastic dynamics that accurately predicts our experience of measurement outcomes. On the other hand, some suggest that we maintain the unitary dynamics of the quantum state, but that certain quantities (e.g. position of particles) have values even though these values are not specified by the quantum state.

Both approaches – the approach that alters the dynamics, and the approach with additional values – are completely successful as responses to the measurement problem in nonrelativistic QM. But both approaches run into obstacles when it comes to synthesizing quantum mechanics with relativity. In particular, the additional values approach (e.g. the de Broglie–Bohm pilot-wave theory) appears to require a preferred frame of reference to define the dynamics of the additional values, and in this case it would fail the test of Lorentz invariance.

The “modal” interpretation of quantum mechanics is similar in spirit to the de Broglie–Bohm theory, but begins from a more abstract perspective on the question of assigning definite values to some observables. Rather than making an intuitively physically motivated choice of the determinate values (e.g. particle positions), the modal interpretation makes the mathematically motivated choice of the spectral decomposition of the quantum state (i.e. the density operator) as determinate.

Unlike the de Broglie–Bohm theory, it is not obvious that the modal interpretation must violate the spirit or letter of relativistic constraints, e.g. Lorentz invariance. So, it seems that there should be some hope of developing a modal interpretation within the framework of Algebraic Quantum Field Theory…..