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Geometric Algebra 3 Dirac Theory and Multiparticle Systems
Chris Doran
Astrophysics Group Cavendish Laboratory Cambridge, UK


Dirac Algebra
· Dirac matrix operators are

· These act on 4-component Dirac spinors

· These spinors satisfy a first-order wave equation

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STA Form
· Adapt the map for Pauli spinors

· Action of the various operators now Imaginary structure still a bivector Dirac equation
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Comments
· Dirac equation based on the spacetime vector derivative · Same as the Maxwell equation, so similar propagator structure · Electromagnetic coupling from gauge principal · Plane wave states have A boost plus a rotation
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Observables
· Observables are 1. Gauge invariant 2. Transform covariantly under Lorentz group

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Current
· Main observable is the Dirac current · Satisfies the conservation equation · Understand the observable better by writing

A Lorentz transformation
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Current 2
· The Dirac current has a wider symmetry group than U(1). · Take · Require · Four generators satisfy this requirement · Arbitrary transform SU(2)
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U(1)
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Streamlines
· The conserved current tells us where the probability density flows · Makes sense to plot current streamlines · These are genuine, local observables · Not the same as following a Bohmian interpretation · No need to insist that a `particle' actually follows a given streamline · Tunnelling is a good illustration
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Tunnelling

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Streamlines
Only front of the packet gets through

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Spin Vector
· The Dirac spin observable is · Same structure as used in classical model · Use a 1D wavepacket to simulate a spin measurement · Magnetic field simulated by a delta function shock · Splits the initial packet into 2

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Streamlines

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Spin Orientation

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Multiparticle Space
· Now suppose we want to describe n particles. · View their trajectories as a path in 4n dimensional configuration space · The vectors generators of this space satisfy

· Generators from different spaces anticommute · These give a means of projecting out individual particle species
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N-Particle Bivectors
· Now form the relative bivectors from separate spaces a a a i i0 · These satisfy

· Bivectors from different spaces commute · This is the GA implementation of the tensor product · `Explains' the nature of multiparticle Hilbert space
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Complex Structure
· In quantum theory, states all share a single complex structure. · So in GA, 2 particle quantum states must satisfy · Define the 2 particle correlator

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2-Particle States
· Correlator ensures that 2-particle states have 8 real degrees of freedom · A 2-particle direct-product state is · Action of imaginary is

· Complex structure now generated by J

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Operators
· Action of 2-particle Pauli operators

· Inner product · Examples

Generic symbol for complex structure

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Density Matrices
· A normalised 2 particle density matrix can be expressed as

· So, for example · All of the information in the density matrix is held in the observables

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Inner products and traces
· Can write the overlap probability as · So have, for n-particle pure states · The partial trace operation corresponds to forming the observables, and throwing out terms · Clearly see how this is removing information · For mixed states, can correlate on pseudoscalar
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Schmidt Decomposition
· General way to handle 2-particle states is to write as a matrix and perform an SVD · Result is a decomposition of the form

· Only works for bipartite states
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GA Form
· Define the local states / operators · Result of the Schmidt decomposition can now be written

Local unitaries

Entangling term

· Now have a form which generalises to arbitrary particles
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3-Particle States
· The GHZ state is · GA equivalent · The W-state is more interesting

· Goes to

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Singlet State
· An example of an entangled,or non-local, state is the 2-particle singlet state

· This satisfies the identity · Gives straightforward proof of invariance · Observables are
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Relativistic States
· All of the previous considerations extend immediately to relativistic states · Can give physical definitions of entanglement for Dirac states · Some disagreement on these issues in current literature · Has been suggested that relative observers disagree on entanglement and purity · More likely that an inappropriate definition has been adopted
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Relativistic Singlet
· Can extend the non-relativistic state to one invariant under boosts as well · This satisfies A Lorentz rotor · This state plays an important role in GA versions of 2-spinor calculus and twistor theory
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Multiparticle Dirac Equation
· Relativistic multiparticle quantum theory is a slippery subject! · Ultimately, most issues sorted by QFT · Can make some progress, though, e.g. with Pauli principle

· Antisymmetrised state constructed via

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Current
· For equal mass particles, basic equation is · Get a conserved current in 8D space · Pauli principle ensures that · Ensures that if 2 streamlines ever met, they could never separate

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Plots

Spins aligned

Spins anti-aligned

In both cases the packets pass through each other
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Exercises
· Verify that the overlap probability between 2 states is

· Now suppose that one state is the singlet, and the other is separable. Prove that

Angle between the spin vectors, or between measuring apparatus
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