Algebra of physical space
In physics, the algebra of physical space (APS) is the use of the Clifford or geometric algebra Cl3,0(R) of the three-dimensional Euclidean space as a model for (3+1)-dimensional spacetime, representing a point in spacetime via a paravector (3-dimensional vector plus a 1-dimensional scalar).
The Clifford algebra Cl3,0(R) has a faithful representation, generated by Pauli matrices, on the spin representation C2; further, Cl3,0(R) is isomorphic to the even subalgebra Cl
3,1(R) of the Clifford algebra Cl3,1(R).
APS can be used to construct a compact, unified and geometrical formalism for both classical and quantum mechanics.
Spacetime position paravector
where the time is given by the scalar part x0 = t, and e1, e2, e3 are the standard basis for position space. Throughout, units such that c = 1 are used, called natural units. In the Pauli matrix representation, the unit basis vectors are replaced by the Pauli matrices and the scalar part by the identity matrix. This means that the Pauli matrix representation of the space-time position is
Lorentz transformations and rotors
The restricted Lorentz transformations that preserve the direction of time and include rotations and boosts can be performed by an exponentiation of the spacetime rotation biparavector W
In the matrix representation the Lorentz rotor is seen to form an instance of the SL(2,C) group (special linear group of degree 2 over the complex numbers), which is the double cover of the Lorentz group. The unimodularity of the Lorentz rotor is translated in the following condition in terms of the product of the Lorentz rotor with its Clifford conjugation
The unitary element R is called a rotor because this encodes rotations, and the Hermitian element B encodes boosts.
This expression can be brought to a more compact form by defining the ordinary velocity as
and recalling the definition of the gamma factor:
so that the proper velocity is more compactly:
The proper velocity transforms under the action of the Lorentz rotor L as
The electromagnetic field, potential and current
The electromagnetic field is represented as a bi-paravector F:
The source of the field F is the electromagnetic four-current:
The field can be split into electric
and F is invariant under a gauge transformation of the form
where is a scalar field.
The electromagnetic field is covariant under Lorentz transformations according to the law
Maxwell's equations and the Lorentz force
The Maxwell equations can be expressed in a single equation:
where the overbar represents the Clifford conjugation.
The Lorentz force equation takes the form
The electromagnetic Lagrangian is
which is a real scalar invariant.
Relativistic quantum mechanics
where e3 is an arbitrary unitary vector, and A is the electromagnetic paravector potential as above. The electromagnetic interaction has been included via minimal coupling in terms of the potential A.
The differential equation of the Lorentz rotor that is consistent with the Lorentz force is
such that the proper velocity is calculated as the Lorentz transformation of the proper velocity at rest
which can be integrated to find the space-time trajectory with the additional use of
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- W. E. Baylis, editor, Clifford (Geometric) Algebra with Applications to Physics And Mathematics Birkhäuser, Boston 1996.
- Chris Doran and Anthony Lasenby, Geometric Algebra for Physicists, Cambridge University Press (2003)
- David Hestenes: New Foundations for Classical Mechanics (Second Edition). ISBN 0-7923-5514-8, Kluwer Academic Publishers (1999)
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- Baylis, W. E.; Yao, Y. (1 July 1999). "Relativistic dynamics of charges in electromagnetic fields: An eigenspinor approach". Physical Review A. American Physical Society (APS). 60 (2): 785–795. doi:10.1103/physreva.60.785. ISSN 1050-2947.