Intuitive Algebra (Groupoid/Categorical Structure) of Open Strings As Morphisms

A geometric Dirichlet brane is a triple (L, E, ∇E) – a submanifold L ⊂ M, carrying a vector bundle E, with connection ∇E.

The real dimension of L is also often brought into the nomenclature, so that one speaks of a Dirichlet p-brane if p = dimRL.

An open string which stretches from a Dirichlet brane (L, E, ∇E) to a Dirichlet brane (K, F, ∇F), is a map X from an interval I ≅ [0,1] to M, such that X(0) ∈ L and X(1) ∈ K. An “open string history” is a map from R into open strings, or equivalently a map from a two-dimensional surface with boundary, say Σ ≡ I × R, to M , such that the two boundaries embed into L and K.

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The quantum theory of these open strings is defined by a functional integral over these histories, with a weight which depends on the connections ∇E and ∇F. It describes the time evolution of an open string state which is a wave function in a Hilbert space HB,B′ labelled by the two choices of brane B = (L, E, ∇E) and B′ = (K, F, ∇F).

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Distinct Dirichlet branes can embed into the same submanifold L. One way to represent this would be to specify the configurations of Dirichlet branes as a set of submanifolds with multiplicity. However, we can also represent this choice by using the choice of bundle E. Thus, a set of N identical branes will be represented by tensoring the bundle E with CN. The connection is also obtained by tensor product. An N-fold copy of the Dirichlet brane (L, E, ∇E) is thus a triple (L, E ⊗CN, ∇E ⊗ idN).

In physics, one visualizes this choice by labelling each open string boundary with a basis vector of CN, which specifies a choice among the N identical branes. These labels are called Chan-Paton factors. One then uses them to constrain the interactions between open strings. If we picture such an interaction as the joining of two open strings to one, the end of the first to the beginning of the second, we require not only the positions of the two ends to agree, but also the Chan-Paton factors. This operation is the intuitive algebra of open strings.

Mathematically, an algebra of open strings can always be tensored with a matrix algebra, in general producing a noncommutative algebra. More generally, if there is more than one possible boundary condition, then, rather than an algebra, it is better to think of this as a groupoid or categorical structure on the boundary conditions and the corresponding open strings. In the language of groupoids, particular open strings are elements of the groupoid, and the composition law is defined only for pairs of open strings with a common boundary. In the categorical language, boundary conditions are objects, and open strings are morphisms. The simplest intuitive argument that a non-trivial choice can be made here is to call upon the general principle that any local deformation of the world-sheet action should be a physically valid choice. In particular, particles in physics can be charged under a gauge field, for example the Maxwell field for an electron, the color Yang-Mills field for a quark, and so on. The wave function for a charged particle is then not complex-valued, but takes values in a bundle E.

Now, the effect of a general connection ∇E is to modify the functional integral by modifying the weight associated to a given history of the particle. Suppose the trajectory of a particle is defined by a map φ : R → M; then a natural functional on trajectories associated with a connection ∇ on M is simply its holonomy along the trajectory, a linear map from E|φ(t1) to E|φ(t2). The functional integral is now defined physically as a sum over trajectories with this holonomy included in the weight.

The simplest way to generalize this to a string is to consider the ls → 0 limit. Now the constraint of finiteness of energy is satisfied only by a string of vanishingly small length, effectively a particle. In this limit, both ends of the string map to the same point, which must therefore lie on L ∩ K.

The upshot is that, in this limit, the wave function of an open string between Dirichlet branes (L, E, ∇) and (K, F, ∇F) transforms as a section of E ⊠ F over L ∩ K, with the natural connection on the direct product. In the special case of (L, E, ∇E) ≅ (K, F, ∇F), this reduces to the statement that an open string state is a section of EndE. Open string states are sections of a graded vector bundle End E ⊗ Λ•T∗L, the degree-1 part of which corresponds to infinitesimal deformations of ∇E. In fact, these open string states are the infinitesimal deformations of ∇E, in the standard sense of quantum field theory, i.e., a single open string is a localized excitation of the field obtained by quantizing the connection ∇E. Similarly, other open string states are sections of the normal bundle of L within X, and are related in the same way to infinitesimal deformations of the submanifold. These relations, and their generalizations to open strings stretched between Dirichlet branes, define the physical sense in which the particular set of Dirichlet branes associated to a specified background X can be deduced from string theory.

Conjuncted: Gauge Theory

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Weyl introduced as a phase factor an exponential in which the phase α is preceded by the imaginary unit i, e.g., e+iqα(x), in the wave function for the wave equations (for instance, the Dirac equation is (iγμμ − m)ψ = 0). It is here that Weyl correctly formulated gauge theory as a symmetry principle from which electromagnetism could be derived. It had been shown that for a quantum theory of charged particles interacting with the electromagnetic field, invariance under a gauge transformation of the potentials required multiplication of the wave function by the now well-know phase factor. Yang cited Weyl’s gauge theory results as reported by Pauli as a source for Yang-Mills gauge theory; although Yang didn’t find out until much later that these were Weyl’s results. Moreover, Pauli did not explicitly mention Weyl’s geometric interpretation. It was only much after Yang and Mills published their article that Yang realized the connection between their work and geometry. Yang says

Whitehead’s Anti-Substantivilism, or Process & Reality as a Cosmology to-be. Thought of the Day 39.0

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Treating “stuff” as some kind of metaphysical primitive is mere substantivilism – and fundamentally question-begging. One has replaced an extra-theoretic referent of the wave-function (unless one defers to some quasi-literalist reading of the nature of the stochastic amplitude function ζ[X(t)] as somehow characterizing something akin to being a “density of stuff”, and moreover the logic and probability (Born Rules) must ultimately be obtained from experimentally obtained scattering amplitudes) with something at least as equally mystifying, as the argument against decoherence goes on to show:

In other words, you have a state vector which gives rise to an outcome of a measurement and you cannot understand why this is so according to your theory.

As a response to Platonism, one can likewise read Process and Reality as essentially anti-substantivilist.

Consider, for instance:

Those elements of our experience which stand out clearly and distinctly [giving rise to our substantial intuitions] in our consciousness are not its basic facts, [but] they are . . . late derivatives in the concrescence of an experiencing subject. . . .Neglect of this law [implies that] . . . [e]xperience has been explained in a thoroughly topsy-turvy fashion, the wrong end first (161).

To function as an object is to be a determinant of the definiteness of an actual occurrence [occasion] (243).

The phenomenological ontology offered in Process and Reality is richly nuanced (including metaphysical primitives such as prehensions, occasions, and their respectively derivative notions such as causal efficacy, presentational immediacy, nexus, etc.). None of these suggest metaphysical notions of substance (i.e., independently existing subjects) as a primitive. The case can perhaps be made concerning the discussion of eternal objects, but such notions as discussed vis-à-vis the process of concrescence are obviously not metaphysically primitive notions. Certainly these metaphysical primitives conform in a more nuanced and articulated manner to aspects of process ontology. “Embedding” – as the notion of emergence is a crucial constituent in the information-theoretic, quantum-topological, and geometric accounts. Moreover, concerning the issue of relativistic covariance, it is to be regarded that Process and Reality is really a sketch of a cosmology-to-be . . . [in the spirit of ] Kant [who] built on the obsolete ideas of space, time, and matter of Euclid and Newton. Whitehead set out to suggest what a philosophical cosmology might be that builds on Newton.

Quantum Geometrodynamics and Emergence of Time in Quantum Gravity

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It is clear that, like quantum geometrodynamics, the functional integral approach makes fundamental use of a manifold. This means not just that it uses mathematical continua, such as the real numbers (to represent the values of coordinates, or physical quantities); it also postulates a 4-dimensional manifold M as an ‘arena for physical events’. However, its treatment of this manifold is very different from the treatment of spacetime in general relativity in so far as it has a Euclidean, not Lorentzian metric (which, apart from anything else, makes the use of the word ‘event’ distinctly problematic). Also, we may wish to make a summation over different such manifolds, it is in general necessary to consider complex metrics in the functional integral (so that the ‘distance squared’ between two spacetime points can be a complex number), whereas classical general relativity uses only real metrics.

Thus one might think that the manifold (or manifolds!) does not (do not) deserve the name ‘spacetime’. But what is in a name?! Let us in any case now ask how spacetime as understood in present-day physics could emerge from the above use of Riemannian manifolds M, perhaps taken together with other theoretical structures.

In particular: if we choose to specify the boundary conditions using the no-boundary proposal, this means that we take only those saddle-points of the action as contributors (to the semi-classical approximation of the wave function) that correspond to solutions of the Einstein field equations on a compact manifold M with a single boundary Σ and that induce the given values h and φ0 on Σ.

In this way, the question of whether the wave function defined by the functional integral is well approximated by this semi-classical approximation (and thus whether it predicts classical spacetime) turns out to be a question of choosing a contour of integration C in the space of complex spacetime metrics. For the approximation to be valid, we must be able to distort the contour C into a steepest-descents contour that passes through one or more of these stationary points and elsewhere follows a contour along which |e−I| decreases as rapidly as possible away from these stationary points. The wave function is then given by:

Ψ[h, φ0, Σ] ≈ ∑p e−Ip/ ̄h

where Ip are the stationary points of the action through which the contour passes, corresponding to classical solutions of the field equations satisfying the given boundary conditions. Although in general the integral defining the wave function will have many saddle-points, typically there is only a small number of saddle-points making the dominant contribution to the path integral.

For generic boundary conditions, no real Euclidean solutions to the classical Einstein field equations exist. Instead we have complex classical solutions, with a complex action. This accords with the account of the emergence of time via the semiclassical limit in quantum geometrodynamics.

On the Emergence of Time in Quantum Gravity

Feynman Path Integrals, Trajectories and Copenhagen Interpretation. Note Quote.

As the trajectory exists by precept in the trajectory representation, there is no need for Copenhagen’s collapse of the wave function. The trajectory representation can describe an individual particle. On the other hand, Copenhagen describes an ensemble of particles while only rendering probabilities for individual particles.

The trajectory representation renders microstates of the Schrödinger’s wave function for the bound state problem. Each microstate by the equation

ψ = (2m)1/4cos(W/h ̄)/(W′)1/2[a − c2/(4b)]1/2

(aφ2 + bθ2 + cφθ)1/2/[a − c2/(4b)]1/2 cos[arctan(b(θ/φ) + c/2)/(ab − c2/4)1/2 = φ

is sufficient by itself to determine the Schrödinger’s wave function. Thus, the existence of microstates is a counter example refuting the Copenhagen assertion that the Schrödinger’s wave function be an exhaustive description of non-relativistic quantum phenomenon. The trajectory representation is deterministic. We can now identify a trajectory and its corresponding Schrödinger wave function with sub-barrier energy that tunnels through the barrier with certainty. Hence, tunneling with certainty is a counter example refuting Born’s postulate of the Copenhagen interpretation that attributes a probability amplitude to the Schrödinger’s wave function. As the trajectory representation is deterministic and does not need ψ, much less to assign a probability amplitude to it, the trajectory representation does not need a wave packet to describe or localize a particle. The equation of motion,

t − τ = ∂W/∂E, where t is the trajectory time, relative to its constant coordinate τ, and given as a function of x;

for a particle (monochromatic wave) has been shown to be consistent with the group velocity of the wave packet. Normalization, as previously noted herein, is determined by the nonlinearity of the generalized Hamilton-Jacobi equation for the trajectory representation and for the Copenhagen interpretation by the probability of finding the particle in space being unity. Though probability is not needed for tunneling through a barrier, the trajectory interpretation for tunneling is still consistent with the Schrödinger representation without the Copenhagen interpretation. The incident wave with compound spatial modulation of amplitude and phase for the trajectory representation,

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has only two spectral components which are the incident and reflected unmodulated waves of the Schrödinger representation.

Trajectories differ with Feynman’s path integrals in three ways. First, trajectories employ a quantum Hamilton’s characteristic function while a path integral is based upon a classical Hamilton’s characteristic function. Second, the quantum Hamilton’s characteristic function is determined uniquely by the initial values of the quantum stationary Hamilton-Jacobi equation, while path integrals are democratic summing over all possible classical paths to determine Feynman’s amplitude. While path integrals need an infinite number of constants of the motion even for a single particle in one dimension, motion in the trajectory representation for a finite number of particles in finite dimensions is always determined by only a finite number of constants of the motion. Third, trajectories are well defined in classically forbidden regions where path integrals are not defined by precept.

Heisenberg’s uncertainty principle shall remain premature as long as Copenhagen uses an insufficient subset of initial conditions (x, p) to describe quantum phenomena. Bohr’s complementarity postulates that the wave-particle duality be resolved consistent with the measuring instrument’s specific properties.

Heisenberg’s uncertainty principle shall remain premature as long as Copenhagen uses an insufficient subset of initial conditions (x, p) to describe quantum phenomena. Bohr’s complementarity postulates that the wave-particle duality be resolved consistent with the measuring instrument’s specific properties. Anonymous referees of the Copenhagen school have had reservations concerning the representation of the incident modulated wave as represented by the equation

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before the barrier. They have reported that compoundly modulated wave represented by the above equation is only a clever superposition of the incident and reflected unmodulated plane waves. They have concluded that synthesizing a running wave with compound spatial modulation from its spectral components is nonphysical because it would spontaneously split. By the superposition principle of linear differential equations, the spectral components may be used to synthesize a new pair of independent solutions with compound modulations running in opposite directions. Likewise, an unmodulated plane wave running in one direction can be synthesized from two waves with compound modulation running in the opposite directions for mappings under the superposition principle are reversible.