What is Clifford algebra: Definition and 32 Discussions

In mathematics, a Clifford algebra is an algebra generated by a vector space with a quadratic form, and is a unital associative algebra. As K-algebras, they generalize the real numbers, complex numbers, quaternions and several other hypercomplex number systems. The theory of Clifford algebras is intimately connected with the theory of quadratic forms and orthogonal transformations. Clifford algebras have important applications in a variety of fields including geometry, theoretical physics and digital image processing. They are named after the English mathematician William Kingdon Clifford.
The most familiar Clifford algebras, the orthogonal Clifford algebras, are also referred to as (pseudo-)Riemannian Clifford algebras, as distinct from symplectic Clifford algebras.

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  1. jv07cs

    I Definition of Clifford Algebras

    I was reading D.J.H. Garling's book "Clifford Algebra: An Introduction" and it defines clifford algebras as follows: But if ##1 \notin j(E)##, how come ##j(E)## generate ##A## since it doesn't generate its identity element?
  2. S

    Quantum Clifford Algebra for Quantum Field Theory, Supersymmetry, Supergravity

    I'm currently trying to learn Clifford algebra or more specifically spinors, in higher dimensions. My goal is to study AdS/CFT, but an essential part of learning it is to understand SUSY which then needs some element of Clifford algebra in higher dimensions. I have consulted, Introduction to...
  3. K

    I ##\mathbb{C}\oplus\mathbb{C}\cong\mathbb{C}\otimes\mathbb{C}##

    Hello! Reading book o Clifford algebra authors claim that ##\mathbb{C}\oplus\mathbb{C}\cong\mathbb{C}\otimes_{\mathbb{R}}\mathbb{C}## as algebras. Unfortunately proof is absent and provided hint is pretty misleading As vector spaces they are obviously isomorphic since ##\dim_{\mathbb{R}}...
  4. W

    Exploring Geometric Algebra and Calculus: Unifying Concepts in Math and Physics

    My primary interesting in joining this group is to facilitate me wrapping my head around geometric algebra (aka Clifford algebra), geometric calculus and its usage in math, physics, and related disciplines. I cannot help but suspect others might be similarly interested. Being of a certain age...
  5. Ken Gallock

    Clifford algebra in higher dimensions

    Homework Statement Consider gamma matrices ##\gamma^0, \gamma^1, \gamma^2, \gamma^3## in 4-dimension. These gamma matrices satisfy the anti-commutation relation $$ \{ \gamma^\mu , \gamma^\nu \}=2\eta^{\mu \nu}.~~~(\eta^{\mu\nu}=diag(+1,-1,-1,-1)) $$ Define ##\Gamma^{0\pm}, \Gamma^{1\pm}## as...
  6. Phylosopher

    I Clifford Algebra and its contributions to physics

    Hello, I see a lot of people enthusiastic about Clifford algebra and its future role for physics, yet I also see a lot of people frustrated opinions about it. Contents in the internet seems really really small compared with other mathematical topics, not to mention less books about it. All...
  7. micromass

    Insights Interview with a Physicist: David Hestenes - Comments

    micromass submitted a new PF Insights post Interview with a physicist: David Hestenes Continue reading the Original PF Insights Post.
  8. Traruh Synred

    A Clifford Algebra and generalizing Dirac equaution

    http://arxiv.org/abs/1001.2485 --- The above paper is about a possible two-time formulation of physics. It is by serious people. To understand it I'm trying to generalized the Dirac eqn. to 3+2 dimensions with signature (++---) I found the following (now closed post) useful...
  9. V

    How do I simplify the calculation of this trace involving six gamma matrices?

    Trace of six gamma matrices I need to calculate this expression: $$Tr(\gamma^{\mu}\gamma^{\nu}\gamma^{\rho}\gamma^{\sigma}\gamma^{\alpha}\gamma^{\beta}\gamma^{5}) $$ I know that I can express this as: $$...
  10. Elemental

    Spin Operators: Axial for QM, Polar in Clifford Algebra?

    Hello folks! New to this forum, so hoping I'm not retreading old ground. The Pauli matrices are spin angular momentum operators in quantum mechanics and thus are axial vectors. But in Clifford algebra in three dimensions they are odd basis elements and thus polar vectors. Hestenes, Baylis, other...
  11. T

    Dirac equation and clifford algebra

    Is it a must to know clifford algebra in order to derive the dirac equation? I recently watch drphysics video on deriving dirac equation and he use two waves moving in opposite directions to derive it, without touching clifford algebra. If this possible, what is the intuition behind it?
  12. G

    Solutions to first order equation with Clifford algebra elements

    In the same way one can show that \nabla^{2}\theta=0 has only one smooth solution, namely \theta=0, I would like to show that \gamma^{i}\partial_{i}\epsilon=0 has only one smooth solution, where \gamma^{i} is a Dirac gamma matrix (or an element of the Clifford algebra), and \epsilon is a...
  13. D

    Quaternions and Clifford Algebra problems

    Hello, I have some problems with understanding some concepts in Quaternions and Clifford Algebra. For example, where can I learn the basic construcion of Clifford Algebra? I'm listing the equalities I did not understand and I appreciate it if you can help me with understanding these : Homework...
  14. R

    Understanding Traceless Proof for Gamma Matrices

    I'm reading through some lecture notes and there is a proof that the gamma matrices are traceless that I've never seen before (I've seen the "identity 0" on wikipedia proof) and I can't work out some of the steps: \begin{align*} 2\eta_{\mu\nu}Tr(\gamma_\lambda) &=...
  15. J

    Help with clifford algebra vector identity

    Homework Statement This is question 1.1 from section 2-1 of New Foundations of Classical Mechanics: Establish the following "vector identities": (a\wedge b) \cdot (c \wedge d) = b\cdot ca \cdot d - b\cdot da \cdot c = b\cdot(c\wedge d)\cdot a Homework Equations The Attempt at...
  16. Q

    How Is Clifford Algebra Applied in Particle Physics and Field Theory?

    Hello, My courses in particle physics and filed theory use several notations in clifford algebra which I have never met before. Could anyone provides me some useful books for clifford algebra in physics?
  17. K

    Regarding the Clifford algebra and spinors

    Hello! I´m currently taking a course in RQM and have some questions for which I didnt get any satisfactory answers on the lecture. All comments are appricieted! 1. Is the gamma zero tensor some kind of metric in the space for spinors? When normalizing our solution to the Dirac equation it...
  18. D

    Solving Clifford Algebra Equation - Need Help!

    I was trying to solve the following equation: \bigwedge\limits_{j=1}^{k}\begin{bmatrix} a_{1,j}\\ a_{2,j}\\ :\\ .\\ a_{k+1,j} \end{bmatrix} Does anyone know how I can solve it? Thanks in advance.
  19. M

    DG - Clifford Algebra / Differential Forms

    Hello Everyone, I'm currently working through a differential geometry book that uses Clifford's algebra instead of differential forms. If anybody has knowledge of both, would you please explain what the differences between the approaches are, and what (if any) are the advantages of each...
  20. F

    Does Clifford algebra solve both QFT and GR

    I noticed a few sources that seem to indicate that Clifford algebra may be used in both QFT and GR. I've seen where the Clifford algebra is a type of associative algebra that generalizes the real numbers, complex numbers, quaternions, and octonions, see Wikipedia on Clifford Algebra. And I've...
  21. C

    Exploring Irreducible Representations of Clifford Algebra

    I'm doing a course which assumes knowledge of Group Theory - unfortunately I don't have very much. Can someone please explain this statement to me (particularly the bits in bold): "there is only one non-trivial irreducible representation of the Cliford algebra, up to conjugacy" FYI The...
  22. C

    Why are spinors interesting, from a Clifford algebra perspective

    Hi, I'm trying to understand spinors better, and I seem to be getting stuck on understanding the reason they're said to transform differently from vectors, and I'd appreciate any help with a justification for that. I'm sure I'm missing something pretty simple, but here goes; Here's what I've...
  23. P

    Clifford Algebra: Representing Vectors & Projecting to Euclidean

    I've read a number of tutorials on Clifford algebra, but I am still unsure of some elementary concepts. For starters, how would I represent a vector in a 2D vector field as a Clifford multivector? For 2D, a multivector is given by A = a0*1 + a1*e1+a2*e2 + a3*e1e2, where 1 is a scalar...
  24. B

    Relation between Clifford algebra and Lorentz algebra

    If the generators of the Lorentz spinor transformation can be expressed in terms of the gamma matrices (which can be used ot build a Clifford algebra), why can't the generators of the Lorentz vector transformation similarly be expressed in terms of the gamma matrices? And why does there exists...
  25. W

    Getting started with Clifford Algebra

    So what background should I have to get started in Clifford Algebra?
  26. N

    Spinors in d dimensions and Clifford algebra

    I bought a book on susy and there is a chapter on spinors in d-dimensions. Now, maybe I am extremely dumb but I just can't understand the first few lines! EDIT: I was being very dumb except that I think there is a typo...See below... BEGINNING OF QUOTE Consider a d-dimensional...
  27. P

    Clifford algebra isomorphic to tensor algebra or exterior algebra?

    Unfortunately there seems to be a misprint in the paper I'm reading which is an introduction to clifford algebra, it says:(I highlighted in red possible misprint, either one of them has to be true misprint if you know what I mean) The Clifford algebra C(V) is isomorphic to the tensor algebra...
  28. garrett

    Explore Geometry of Symmetric Spaces & Lie Groups on PF

    A few friends have expressed an interest in exploring the geometry of symmetric spaces and Lie groups as they appear in several approaches to describing our universe. Rather than do this over email, I've decided to bring the discussion to PF, where we may draw from the combined wisdom of its...
  29. CarlB

    How Does Clifford Algebra Apply to Quantum Physics?

    I registered the website http://www.CliffordAlgebra.com . My purpose is to have a website that gives a decent education in the practical uses of Clifford Algebra, as I am interested in their applications to physics. Is there any desire over here for a more mathematical introduction to the...
  30. A

    Proving No Non-Trivial Ideals in Clifford-t Algebra for t ≠ 0 | V = W ⊕ W*

    Let \mathbb{K} be a field. Assume that any vector spaces mentioned hereafter have \mathbb{K} underlying them. W is a vector space, and W* is the dual space. If Y, Z are vector spaces, define the vector space: Y * Z = Span{F(y,z) | y in Y, z in Z} So if (y,z) and (y',z') are distinct...
  31. T

    Clifford Algebra Question on Vectors

    I'm a Software Developer by profession, not a Mathematician, or Physicist, so please be patient with my ignorance as I'm about to ask (what I am sure is) a very basic question about Clifford Algebra. I've been reading some Clifford Algebra books I have, on how C.A. represents and performs...