Does Clifford algebra solve both QFT and GR

In summary, the conversation discusses the potential use of Clifford algebra in both quantum field theory and general relativity. It is noted that the algebra generalizes real numbers, complex numbers, quaternions, and octonions and has been used in the description of symmetries in the standard model. It is also mentioned that the algebra can be used in an alternative description of differential geometry in the formulation of GR. The speaker asks if this common algebra can be used to derive either GR or QFT, and if it is accepted as a fundamental tool in physics. They also inquire about resources for learning more about quaternions, octonions, and Clifford algebra and how they are used in the standard model and GR. The response suggests that a
  • #1
friend
1,452
9
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 seen where the complex numbers, quaternions, and octernions can be used in the description for the U(1), SU(2), SU(3) symmetries of the Standard Model, see this article, and this book. However, I've also seen where the Clifford algebra can be used in an alternative description of differential geometry used in the formulation of GR, see this book for example.

So my question is does this common algebra allow us to derive GR in terms of QFT or visa versa? Or if we could justify the use of the complex numbers, quaternions, octonions, and the Clifford algebra by some other means, could we derive both QFT and GR from that common justification of the algebra? What more information or constraints would be needed to do so?
 
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  • #2
I'm trying to decide whether I should research the Clifford algebra and how it applies to physics. I'm looking to get a sense of how valid is the application of Clifford algebra in physics. Is there a consenses that Clifford algebra, quaternions, and octonions are indeed fundamental or at least have a valuable use?

If the use of these algebras is valid in physics, then I have to wonder... Since the Clifford algebra is used in differential geometry, and quaterions and octonions are used in the SM, then can the quaternions and octonions be used to contrain the Clifford algebra in such a way in differential geometry as to product GR if the SM is assumed? Or is it more the case that the Clifford algebra is used on the metric whereas the quaternion and octonions are used on wavefunctions, which is a totally different animal? Any insight would be appreciated. Thanks.
 
  • #3
friend said:
I'm trying to decide whether I should research the Clifford algebra and how it applies to physics...QUOTE]

What a strange question! Of course Clifford (geometric) algebra does apply to physics! Just think about spinors, Dirac equation...
 
  • #4
Blackforest said:
What a strange question! Of course Clifford (geometric) algebra does apply to physics! Just think about spinors, Dirac equation...

Yes, some people are using it as the basis of the SM. But I'm having trouble getting started with my studies of quaternions, octonions, and clifford algebra. My linear algebra and group theory is weak. Do you know of any easy introductions to quaternions, octonions, and clifford algebra?

So when they use the quaternions and octonions in the SM, do they use the algebra to establish commutation relation between fields? How is the Clifford algebra used in GR? Is it used on connection fields perhaps? Does it establish commutation relations, or perhaps the Bianchi identity, what? Thanks.
 
  • #5
Clifford algebras are particularly necessary for understanding spinor representations and spin groups. Depending on what you're interested in studying, they could be important (a good knowledge of the mathematical underpinnings of SM and QFT will require knowledge of Clifford algebras). I'd say that quaternions (and octonions) are less important: quaternions offer an alternative to the tensor formulation of GR, and are not something I'd concern myself with on an initial read through.

I would certainly get up to speed on your linear algebra and group theory (in particular Lie groups and Lie algebras which are widespread in particle physics) before I'd worry about Clifford algebras and quaternions.
 
  • #6
bapowell said:
Clifford algebras are particularly necessary for understanding spinor representations and spin groups. Depending on what you're interested in studying, they could be important (a good knowledge of the mathematical underpinnings of SM and QFT will require knowledge of Clifford algebras). I'd say that quaternions (and octonions) are less important: quaternions offer an alternative to the tensor formulation of GR, and are not something I'd concern myself with on an initial read through.

I would certainly get up to speed on your linear algebra and group theory (in particular Lie groups and Lie algebras which are widespread in particle physics) before I'd worry about Clifford algebras and quaternions.

The problem is that I seem to have been able to justify the use of quaternions and octonions from purely logical reasons. And now I wonder if this is a road into physics. So I feel like I need to go from these hypercomplex numbers to SM physics. You seem to indicate that you can go from the SM to hypercomplex numbers. But can one go from hypercomplex numbers to the SM?
 
  • #7
friend said:
Yes, some people are using it as the basis of the SM. But I'm having trouble getting started with my studies of quaternions, octonions, and clifford algebra. My linear algebra and group theory is weak. Do you know of any easy introductions to quaternions, octonions, and clifford algebra?

So when they use the quaternions and octonions in the SM, do they use the algebra to establish commutation relation between fields? How is the Clifford algebra used in GR? Is it used on connection fields perhaps? Does it establish commutation relations, or perhaps the Bianchi identity, what? Thanks.

If with SM you mean standard model, then yes some parts of the Lagrangian are involving the Pauli matrices and you can consider that these matrices are a representation of the quaternions or of a Clifford algebra. See the theory of spinors by E. Cartan himself.
 
  • #8
Blackforest said:
If with SM you mean standard model, then yes some parts of the Lagrangian are involving the Pauli matrices and you can consider that these matrices are a representation of the quaternions or of a Clifford algebra. See the theory of spinors by E. Cartan himself.

I hope it is allowed to propose a (commercial) link here but if you need a reference either type the words on google or go to the following link (but it is not the only possible one): https://www.amazon.com/dp/0486640701/?tag=pfamazon01-20

You get everything you need to beginn with and if you are patient, at the very end of the book, a part of the answer to your initial question.
 
  • #9
Blackforest said:
I hope it is allowed to propose a (commercial) link here but if you need a reference either type the words on google or go to the following link (but it is not the only possible one): https://www.amazon.com/dp/0486640701/?tag=pfamazon01-20

You get everything you need to beginn with and if you are patient, at the very end of the book, a part of the answer to your initial question.

Yes, the book is on its way. Thanks.

But from what I can tell from what I've seen on the Net, clifford algebra is used either for geometry or for the SM in the form of quaternions and octonions. What I don't see is anyone connecting the two. Is this because the algebra operates on two different objects that can't be connected in some way?
 
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  • #10
For example, is there a connection between spin foam networks of LQG and the spinors derived using clifford algebra?
 
  • #11
friend said:
For example, is there a connection between spin foam networks of LQG and the spinors derived using clifford algebra?

It is far far far over my head and I cannot help you or explain details.

But I just guess that the answer is: some where yes.

Good luck

http://arxiv.org/pdf/1201.2120v1.pdf
 
  • #12
Relevant References

For Clifford/Geometric Algebra the most relevant reference is -

"Geometric Algebra for Physicsts" by Doran and Lasenby

https://www.amazon.com/dp/0521715954/?tag=pfamazon01-20

Which covers (among other things) both the Dirac equation and the Gauge Theory of Gravity.

I have a set of notes based on Doran and Lasenby and symbolic software in python at

https://github.com/brombo/GA
 
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  • #13
What I'm reading so far is that the S0(1,3) symmetry of Lorentz invariance is a "double cover" for the SU(2) symmetry of the Standard Model. And I'm read other places that the spin of a particle is deeply connected to Lorentz invariance. Is this deep connection the double cover?

What does double cover mean anyway? Is this covered in the book you mention? Thanks.
 
  • #14
Double Cover in Doran and Lasenby

It is mentioned in D&L. In GA rotations (in any dimension) are preformed with rotors which are scalars plus bivectors, R, with the condition that R*rev(R) = 1 (rev(R) is the reverse of R and * the geometric product). Then to rotate a vector, x, in the plane that R defines you evaluate

R*x*rev(R)

Since rev(-R) = -rev(R) if you replace R with -R you get the same result. This is the double cover. Note that R = exp((theta/2)*B) where B is a normalized bivector defining the plane of rotation. For the Lorentz transformation in Minkowski space if B is a spacelike plane if B**2 = -1 and

R = cos(theta/2)+sin(theta/2)*B

If the plane is timelike B**2 = 1 and

R = cosh(theta/2)+sinh(theta/2)*B

also note that rev(R) = exp(-(theta/2)*B). Multiple Lorentz transformation R1, R2 are evaluated with

R2*R1*x*rev(R1)*rev(R2)

and the geometric product * is the group operation.
 
  • #15
Hello friend, there's been lots of good advice here so far. While playing with Clifford algebra is great fun, there's a risk of both overestimating and underestimating its importance in theoretical physics, and it's good to understand its context. Clifford algebra is useful in physics because the antisymmetric Clifford product between Clifford bivector elements is isomorphic to the Lie bracket between corresponding elements of the spin Lie algebra. This means Clifford algebra is hugely important in fundamental physics, but not a replacement for all of differential geometry. One still needs to understand Lie groups and differential forms, and then have even more fun with Clifford valued differential forms, specifically for describing connections (gauge fields). Even more specifically, gravity is well described using a spin connection, which is a spin(1,3)-valued 1-form, which can also be written as a Cl2(1,3)-valued 1-form, which usually acts on a Cl1(1,3)-valued gravitational frame as well as on spinors, which are defined using matrix representations of the spin group which correspond to matrix representations of Clifford algebras. Clifford algebra is also useful in the Standard Model because u(1)xsu(2)xsu(3) embeds in spin(10) (the Lie algebra of the Spin(10) GUT) or equivalently Cl2(10), with the SM gauge fields all parts of a spin(10)-valued 1-form, and the SM fermion multiplets living in 16 dimensional spinor representation spaces. There are, of course, other things one can do with Clifford algebras, including how this structure unifies in spin(11,3) and E8, but that's my opinion on their importance in fundamental physics. Hope you find that helpful.
 
  • #16
Remind me again, U(1) is the symmetry of the Electromagnetic force and is represented by the dirac matrix? And SU(2) is the symmetry of the Weak force and is represented by the Pauli matix, and SU(3) is the symmetry of the Strong force and is represented by the what matix? Am I close?
 

1. What is Clifford algebra?

Clifford algebra is a mathematical framework that extends the properties of vectors and matrices to higher dimensions. It is based on the concept of geometric algebra, which uses geometric objects such as points, lines, and planes to represent mathematical operations.

2. How does Clifford algebra relate to QFT and GR?

Clifford algebra has been used in both quantum field theory (QFT) and general relativity (GR) to provide a unified mathematical framework for these theories. This is because it allows for the representation of both spinors and tensors, which are essential mathematical objects in these theories.

3. Can Clifford algebra solve the problems with QFT and GR?

While Clifford algebra provides a unified framework for QFT and GR, it does not necessarily solve all of the problems associated with these theories. It allows for a more elegant and geometric interpretation of the equations, but it does not necessarily provide new solutions to existing problems.

4. Are there any limitations to using Clifford algebra in QFT and GR?

One limitation of using Clifford algebra in QFT and GR is that it can be a more complex and abstract mathematical framework, making it challenging to apply in practice. Additionally, it may not be suitable for every problem, and other mathematical tools may be more appropriate in certain cases.

5. How can I learn more about Clifford algebra and its applications in QFT and GR?

There are many resources available for learning about Clifford algebra and its applications in QFT and GR. These include books, online courses, and research papers. It is recommended to have a strong foundation in mathematics and physics before delving into this topic.

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