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Geometric Algebra 2 Quantum Theory
Chris Doran
Astrophysics Group Cavendish Laboratory Cambridge, UK


Spin
· Stern-Gerlach tells us that electron wavefunction contains two terms · Describe state in terms of a spinor
S N

· A 2-state system or qubit
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Pauli Matrices
· Operators acting on a spinor must obey angular momentum relations · Get spin operators

· These form a Clifford algebra · A matrix representation of the geometric algebra of 3D space
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Observables
· Want to place Pauli theory in a more geometric framework with · Construct observables · Belong to a unit vector · Written in terms of polar coordinates, find parameterisation

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Rotors and Spinors
· From work on Euler angles, encode degrees of freedom in the rotor · Represent spinor / qubit as element of the even subalgebra:

· Verify that
Keeps result in even algebra
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Imaginary Structure
· Can construct imaginary action from

· So find that · Complex structure controlled by a bivector · Acts on the right, so commutes with operators applied to the left of the spinor · Hints at a geometric substructure · Can always use i to denote the structure
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Inner Products
· The reverse operation in 3D is same as Hermitian conjugation · Real part of inner product is · Follows that full inner product is · The projection onto the 1 and I components,
3

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Observables
· Spin observables become · All information contained in the spin vector · Now define normalised rotor · Operation of forming an observable reduces to Same as classical
expression
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Rotating Spinors
· So have a natural `explanation' for 2-sided construction of quantum observables · Now look at composite rotations · So rotor transformation law is · Take angle through to 2
Sign change for fermions
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Unitary Transformations
· Spinors can transform under the full unitary group U(2) · Decomposes into SU(2) and a U(1) term · SU(2) term becomes a rotor on left · U(1) term applied on the right · Separates out the group structure in a helpful way · Does all generalise to multiparticle setting
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Magnetic Field
· · · · Rotor contained in 1/2 R Use this to Simplify equations Magnetic field Schrodinger equation

· Reduces to simple equation · Magnitude is constant, so left with rotor equation
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Density Matrices
· Mixed states are described by a density matrix · For a pure state this is

· GA version is · Addition is fine in GA! · General mixed state from sum

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Spacetime Algebra
· Basic tool for relativistic physics is the spacetime algebra or STA.

1









I



I 0 1 2
1 pseudoscalar

3

1 scalar 4 vectors 6 bivectors 4 trivectors

· Generators satisfy · A matrix-free representation of Dirac theory · Currently used for classical mechanics, scattering, tunnelling, supersymmetry, gravity and quantum information
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Relative Space
· Determine 3D space relative for observer with velocity given by timelike vector 0 · Suppose event has position x in natural units

· The basis elements of relative vector are · Satisfy

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Relative Split
· Split bivectors with 0 to determine relative split 1 i , I i I I

1

i

I i

I

· Relative vectors generate 3D algebra with same volume element · Relativistic (Dirac) spinors constructed from full 3D algebra
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Lorentz Transformation
· Moving observers construct a new coordinate grid · Both position and time coordinates change
Time

Space

Need to re-express this in terms of vector transformations
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Frames and Boosts
· Vector unaffected by coordinate system, so · Frame vectors related by · Introduce the hyperbolic angle · Transformed vectors now
Exponential of a bivector
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Spacetime Rotors
· Define the spacetime rotor · A Lorentz transformation can now be written in rotor form · Use the tilde for reverse operation in the STA (dagger is frame-dependent) · Same rotor description as 3D · Far superior to 4X4 matrices!
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Pure Boosts
· Rotors generate proper orthochronous transformations · Suppose we want the pure boost from u to v · Solution is

· Remainder of a general rotor is
A 3D rotor

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Velocity and Acceleration
· Write arbitrary 4-velocity as · Acceleration is

· But · So

Pure bivector

Acceleration bivector
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The vector derivative
· Define the vector derivative operator in the standard way

· So components of this are directional derivatives · But now the vector product terms are invertible · Can construct Green's functions for · These are Feynman propagators in spacetime
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2 Dimensions
· Vector derivative is

· Now introduce the scalar+pseudoscalar field · Find that

· Same terms that appear in the CauchyRiemann equations!
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Analytic Functions
· Vector derivative closely related to definition of analytic functions · Statement that is analytic is 0 · Cauchy integral formula provides inverse · This generalises to arbitrary dimensions · Can construct power series in z because · Lose the commutativity in higher dimensions · But this does not worry us now!
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Spacetime Vector Derivative
· Define spacetime vector derivative

· Has a spacetime split of the form · First application - Maxwell equations

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Maxwell Equations
· Assume no magnetisation and polarisation effects and revert to natural units · Maxwell equations become, in GA form

· Naturally assemble equations for the divergence and (bivector) curl · Combine using geometric product

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STA Form
· Define the field strength (Faraday bivector) · And current · All 4 Maxwell equations unite into the single equation · Spacetime vector derivative is invertible, can carry out first-order propagator theory · First-order Green's function for scattering
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Application

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Lorentz Force Law
· Non-relativistic form is

· Can re-express in relativistic form as

· Simplest form is provided by rotor equation

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Spin Dynamics
· Suppose that a particle carries a spin vector s along its trajectory · Simplest form of rotor equation then has

· Non relativistic limit to this equation is Equation for a particle with g=2!
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Exercises
· 2 spin-1/2 states are represented by and , with accompanying spin vectors · Prove that

· Given that · Prove that
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