vector analysis - garretstar.com · clifford algebra to geometric calculus: a unified language for...
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Vector Analysis
Of Spinors
by
Garret Sobczyk
Universidad de Las Americas-P
Cholula, Mexico
October 2014
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Contents
I. Introduction.
a) A bit of history.
b) Complex numbers.
c) Hyperbolic numbers.
II. What is Geometric Algebra?
a) Geometric numbers of the plane.
b) Geometric numbers of 3-space.
c) Cancellation property
d) Reflections and rotations.
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Contents
III. Idempotents and the Riemann sphere
a) What is an Idempotent?
b) Canonical forms for idempotents.
c) The Riemann sphere.
IV. Spinors.
a) What is a spinor?
b) Norm of a spinor.
c) Properties of spinors.
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Contents
V. Magic of Quantum Mechanics.
a) Expected value.
b) Uncertainty principle.
c) Schrodinger equation.
d) Neutrino oscillation.
VI. Selected References
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What is Geometric Algebra?
Geometric algebra is the completion of the real number system to include new anticommuting square roots of plus and minus one, each such root representing an orthogonal direction in successively higher dimensions.
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Complex and Hyperpolic Numbers
u2=1
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Hyperbolic Numbers
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:
Geometric Numbers G2 of the Plane
Standard Basis of
G2={1, e1, e2, e12}.
where i=e12 is a unit
bivector.
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Basic Identities ab =a.b+a^b
a.b=½(ab+ba)
a^b=½(ab-ba)
a2=a.a= |a|2
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Geometric Numbers of 3-Space
a^b=i axb
a^b^c=[a.(bxc)]i
where i=e1e2e3=e123
a.(b^c)=(a.b)c-(a.c)b
= - ax(bxc).
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Cancellation Property
Let a, b, c be vectors in G3, then
provided
This is equivalent to:
implies
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Reflections L(x) and Rotations R(x)
where |a|=|b|=1 and
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The Spectral Basis of the Geometric
Algebra G3 G3=span{1,e1,e2,e3,e12 ,e13 ,e23,e123}.
By the spectral basis of G3 we mean
where
are mutually annihiliating idempotents.
Note that e1 u+ = u- e1.
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For example, if
then the element g Ɛ G3 is
Pauli Matrices:
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Idempotents and the Riemann
sphere
An element is an idempotent if
This implies that
where
and m and n are orthogonal vectors.
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We find that
where
is an idempotent. We also find
for the idempotent
Important property of idempotents:
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The Riemann Sphere
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What is a Spinor?
Pauli Spinor:
Also called a ket-vector.
Geometric Spinor:
with matrix
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Norm of a Spinor
Sesquilinear inner product:
A spinor is normalized if
where
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Properties of Spinors
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Great Circle of Riemann Sphere
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Magic of Quantum Mechanics
Observable:
where
Note that
is an Hermitian matrix.
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Expected Value:
Standard deviation:
Uncertainty Principle for observables S and T:
or
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Schrodinger Equation
If the Hamiltonian is time independent,
then H = S = s0+s; with solution:
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Neutrino Oscillation (two states)
Electron neutrino state:
Muon neutrino state:
Oscillation:
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The neutrino in the evolving state
will be observed in the state
with the probability
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Spacetime Algebra G1,3 We start with
We factor e1, e2, e3 into Dirac bivectors,
where
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Splitting Space and Time
The ordinary rotation
is in the blue plane of
the bivector i=e12. The
blue plane is boosted
into the yellow plane by
with the velocity v/c = Tanh ɸ.
The light cone is shown in red.
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Selected References
R. Ablamowicz, G. Sobczyk, Lectures on Clifford (Geometric) Algebras and Application, Birkhauser, Boston 2004.
T.F. Havel, GEOMETRIC ALGEBRA: Parallel Processing for the Mind (Nuclear Engineering) 2002. http://www.garretstar.com/secciones/
D. Hestenes, New Foundations for Classical Mechanics, 2nd Ed., Kluwer 1999.
D. Hestenes and G. Sobczyk. Clifford Algebra to Geometric Calculus: A Unified Language for Mathematics and Physics, 2nd edition, Kluwer 1992.
P. Lounesto, Clifford Algebras and Spinors, 2nd Edition. Cambridge University Press, Cambridge, 2001.
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P. Lounesto, CLICAL software packet and user manual. Helsinki University of Technology of Mathematics, Research, Report A248, 1994.
G. Sobczyk, The missing spectral basis in algebra and number theory, The American Mathematical Monthly 108 April 2001, pp. 336-346.
G. Sobczyk, Geometric Matrix Algebra, Linear Algebra and its Applications, 429 (2008) 1163-1173.
G. Sobczyk, Hyperbolic Number Plane, The College Mathematics Journal, Vol. 26, No. 4, pp.269-280, September 1995.
G. Sobczyk, Spacetime Vector Analysis, Physics Letters, 84A, 45-49, 1981.
G. Sobczyk, Noncommutative extensions of Number: An Introduction to Clifford's Geometric Algebra, Aportaciones Matematicas Comunicaciones}, 11 (1992) 207-218.
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G. Sobczyk, Hyperbolic Number Plane, The College Mathematics
Journal, 26:4 (1995) 268-280.
G. Sobczyk, The Generalized Spectral Decomposition of a Linear Operator, The College Mathematics Journal, 28:1 (1997) 27-38.
G. Sobczyk, Spectral integral domains in the classroom,
APORTACIONES MATEMATICAS, Serie Comunicaciones Vol. 20,
(1997) 169-188.
G. Sobczyk, New Foundations in Mathematics: The Geometric Concept of Number, Springer-Birkhauser 2012.
Note: Copies of many of my papers can be found on my website:
http://www.garretstar.com/