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This question may fit better in the discussion of "Why Complex variables are required by QM?", but it also relates to the extent to which arguments by Hestenes are accepted in mainstream physics and may deserve its own discussion?

It seems to me that in the paper "The zitterbewegung interpretation of Quantum Mechanics" Foundations of Physics, 20 1213-1232 Prof Hestenes argues that spin is an actual precession of something physical and that complex variables exist in Quantum Mechanics because of this. I have also heard that Geometric Algebra, which I think was invented by Prof Hestenes can clearly explain why complex variables are required in QM. If these are both the case (spin is a physical phenomenon and GA explains why complex variables are used in QM) can anyone explain why QM is not taught beginning with the assumptions of Geometric Algebra?

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  • $\begingroup$ it's called the fundamental theorem of algebra $\endgroup$ Oct 11 '12 at 15:42
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While this is an old question, I imagine many encounter geometric algebra through David Hestenes's writing on zitterbewegung, so I'd like to offer a different perspective than the existing answers.

Independently of Hestenes's zitterbewegung program, his development of Dirac theory provides a clarity to what is being described in the theory that is difficult to accomplish in the usual matrix representation. This is in part due to the geometrical interpretation provided to the imaginary $i$ in the Dirac equation. While I don't know that this interpretation univocally explains the presence of complex numbers in QM, it certainly provides a way to make sense of it.

There are many algebraic objects that square negatively in a geometric algebra, and these generate rotations in distinct planes represented by those objects. So from the geometric algebra point of view, the imaginary $i$ could be thought of as a generator of rotations in a generic plane.

Let's consider a few examples in spacetime, $\mathcal{\text{Cl}}_{1,3}$.

The bivector $\gamma_2 \gamma_1$ squares to $-1$ and generates rotations in the plane spanned by the orthonormal vectors $\gamma_1$ and $\gamma_2$. This acts on vectors in the following way:

$$v \mapsto v' = e^{\gamma_2\gamma_1\phi/2} v e^{-\gamma_2\gamma_1\phi/2}.$$

Or consider the pseudoscalar $I$ which generates rotations between multivectors and their Hodge duals. In electrodynamics, the pseudoscalar generates duality rotations of the Faraday bivector $F$ via: $$F \mapsto F' = e^{I\beta/2} F e^{I \beta/2}.$$

(Yes without the minus — the reason is the general transformation law $M' = \psi M \tilde \psi$ involves reversion $\tilde \psi$ that behaves differently for these two examples: $\widetilde{\gamma_2\gamma_1} = - \gamma_2\gamma_1$ but $\tilde I = I$.)


As an aside (in response to Ron's comment), the symmetric stress-energy tensor in electrodynamics is given by

$$T(n) = \frac{1}{2} F n \tilde F,$$

which represents the flow of energy-momentum through the surface perpendicular to $n$, and describes a reflection of $n$ across the bivector $F$. I personally find this to be clearer and more manageable than:

$$T^{\mu\nu} = F^{\mu\alpha}F^{\nu}{}_{\alpha} - \frac{1}{4} \eta^{\mu\nu}F_{\alpha\beta}F^{\alpha\beta} = \gamma^\mu \cdot T(\gamma^\nu),$$

so I think it's misleading to say that this formalism "has a hard time with symmetric tensors," even if tensors form a larger algebra.


There are also bivectors that square positively and generate hyperbolic rotations (Lorentz boosts) in spacetime. For instance, take the timelike bivector $\gamma_3 \gamma_0$, which squares to $+1$. This boosts vectors in the $\gamma_3$ direction in the "$\gamma_0$-frame" as:

$$v \mapsto v' = e^{\gamma_3\gamma_0\phi/2} v e^{-\gamma_3\gamma_0\phi/2}.$$

General Lorentz transformations are of the form $R = e^{B/2}$ where $B$ is a spacetime bivector, with vectors transforming as

$$v \mapsto v' = R v \tilde R = e^{B/2} v e^{-B/2}.$$

Here's the point. The dynamical quantity of Dirac theory (a spinor) is precisely a dilation, duality, and Lorentz transformation given by

$$\psi = (\rho e^{I \beta})^{1/2} R.$$

The Dirac equation

$$\nabla \psi \gamma_2 \gamma_1 = \psi p_0$$

describes the dynamics of a spinor that rotates, dilates, and boosts the properties of an electron from its rest frame into some other frame.

In particular, $\psi$ transforms the electron's momentum in its rest frame $p_0 = m \gamma_0$ into a conserved probability current density

$$m J = \psi p_0 \tilde \psi = \rho R p_0 \tilde R$$

and its spin $S_0 = \gamma_2 \gamma_1$ (a bivector) into

$$S = \psi S_0 \tilde \psi = \rho R S_0 \tilde R e^{I \beta}.$$

The isomorphism between the above equation and the matrix representation of the Dirac equation

$$i \gamma^\mu \partial_\mu \lvert \psi \rangle = m \lvert \psi \rangle$$

is given by

$$\gamma^\mu \lvert \psi \rangle \leftrightarrow \gamma^\mu \psi \gamma_0$$ $$i \lvert \psi \rangle \leftrightarrow \psi \gamma_2 \gamma_1,$$ $$\gamma_5 \lvert \psi \rangle = -i \gamma_0\gamma_1\gamma_2\gamma_3 \lvert \psi \rangle \leftrightarrow \psi \gamma_3 \gamma_0,$$

such that the spin bivector $S$ is the observable associated with the imaginary $i$ in the matrix representation. (see p279 of Doran and Lasenby's Geometric Algebra for Physicists, a textbook that I recommend if you're interested in digging in)

This allows us to view U(1) gauge transformations as rotations associated with a particular spatial plane, rather than the complex plane, since $S_0$ is the generator of the gauge symmetry of Dirac theory in this context. Namely, observables (e.g. probability current density and spin density) are invariant under spatial rotations in the plane spanned by the vectors $\gamma_1$ and $\gamma_2$, when performed in the electron's rest frame.

This representation of U(1) as rotations in a plane is already implicit in the standard Hilbert space formalism for a single qubit. The particular plane is of course a matter of convention, equivalent to a choice of reference state relative to which all other states are defined. In particular, any state $\lvert \psi \rangle$ can be written $$\lvert \psi \rangle = U_\psi \lvert 0 \rangle.$$

The fact that $\lvert 0 \rangle = \sigma_z \lvert 0 \rangle$ is an eigenstate of the Pauli operator $\sigma_z$ implies that $$e^{i \phi} \lvert \psi \rangle = U_\psi e^{i \sigma_z \phi} \lvert 0 \rangle$$ which shows that U(1) phase transformations are represented in SU(2) by rotations in the spin plane defined by a given reference state — in this case, rotations in the $i \sigma_z$ plane, which in $\mathcal{Cl}_{1,3}$ is represented by $\gamma_2 \gamma_1 = I \sigma_z$.

I certainly find this to be helpful in understanding the role of complex numbers in Dirac theory, so if you find yourself wondering, geometric algebra is a good language to wonder in.

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The reason it is not taught is because geometric algebra (Clifford algebra to the rest of the world) is a specialized tool for producing certain representations of the rotation/Lorentz group, and it does not have a distinguished place as a defining algebra of space-time.

What Hestenes does is reformulate everything using Clifford algebras instead of the usual coordinates. This is like taking the "slash" of every vector. You can do this, but it is difficult to motivate. Hestenes' motivation is to make an algebra out of the space-time coordinates. But it is not at all clear that one should be able to multiply two vectors and get something sensible out. Why should it be so physically? The reason is clear when you have Dirac matrices, but to introduce this as an axiom is unmotivated from a physical point of view, and I do not think will help pedagogically.

Further, this formalism, while it naturally incorporates forms, has a hard time with symmetric tensors, which are just as natural as antisymmetric ones. You can represent symmetric tensors, of course, using spin indices, but there is no reason to prefer the Clifford algebra way, because you can use other formalisms with equivalent content.

To produce an algebra out of vectors and to claim that it is physical, requires an argument that vectors should multiply together to make antisymmetric tensors plus a scalar. This argument is lacking--- the scalar product and the wedge product are two separate ideas which do not need to be combined into a Clifford algebra unless you are motivated by spin-1/2.

I think that this idea is cute, but it is not pedagogically useful in itself. It might be useful as a way of motivating the Ramond construction in string theory (this is just a hunch, I don't know how to do this), or other otherwise mysterious Dirac matrix intuitions.

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  • $\begingroup$ Supersymmetry comes from considering these kind of handy approaches to constructing representations. Is there a sensible way to see/estimate what one might miss if one only takes these specific tools? How to specifiy the compliment of a method to construct objects (represenations) like that? $\endgroup$
    – Nikolaj-K
    Oct 11 '12 at 14:59
  • $\begingroup$ @NickKidman: This is not a handy approach, it's a truncation of Dirac algebra. If you use general tensor methods, then you are fine, but that's not GA. GA truncates, and truncation is always bad. Generality is good. SUSY does not use GA, it uses superspace, and this is a trick that illuminates and obscures in nearly equal measures. One needs to learn the different superspaces along with a component approach to get a handle on it. $\endgroup$
    – Ron Maimon
    Oct 11 '12 at 17:47
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    $\begingroup$ @RonMaimon It is incorrect to say that GA is a truncation of the Dirac algebra. I think you mean to say that Hestenes's Spacetime Algebra ($G_{1,3}$) is a truncation of the Dirac algebra. Truncation is not always bad, and generality is not always good. Truncating redundant degrees of freedom, for instance, is good. Use of $G_{1,3}$ eliminates the 8 unnecessary degrees of freedom from the Dirac equation, and this is good. $\endgroup$
    – Luke Burns
    Nov 27 '16 at 14:59
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    $\begingroup$ Furthermore, it doesn't matter that "it is not at all clear that one should be able to multiply two vectors and get something sensible out." It turns out that you do get something sensible out and that it is useful. $\endgroup$
    – Luke Burns
    Nov 27 '16 at 15:17
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    $\begingroup$ This is an odd answer that shortchanges Clifford algebras. By unifying the treatment of scalars, vectors, rotations/bivectors and oriented volume elements it actually motivates the usage of Pauli and Dirac matrices (as representations of even subalgebras). It took the mind of someone like Dirac to conceive of them in the standard presentation, but with the Clifford algebra approach they are far more natural $\endgroup$
    – nimish
    Nov 21 '18 at 2:06
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Complex numbers certainly appear in QM and are very convenient, but, as I wrote here before, it is not obvious that they (or pairs of real numbers) are necessary. For example, one can make a scalar wavefunction real by a gauge transform (Schrödinger (Nature (London) 169, 538 (1952))). A similar result was obtained for the Dirac equation without any use of geometric algebra (http://akhmeteli.org/wp-content/uploads/2011/08/JMAPAQ528082303_1.pdf (my article published in the Journal of Mathematical Physics) or http://arxiv.org/abs/1008.4828 ).(EDITED)

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Spinors can indeed be interpreted as objects in a real Clifford algebra (see Luke Burns's answer), and I think that the general utility of complex numbers in (classical and quantum) wave mechanics can probably be traced back to the frequent appearance in real Clifford algebras of subalgebras that are isomorphic to the complex numbers.

But as far as I can tell, the complex numbers in spinors are not related to the complex amplitudes of quantum mechanics:

  • Two-component complex Weyl spinors have been used to reformulate general relativity. See e.g. Spinors and Space-Time by Penrose and Rindler. This may have been motivated by a search for a theory of quantum gravity, but it isn't quantum gravity, it's classical gravity with complex spinors.

  • In path-integral quantum field theory, the quantum amplitude for a transition from a particular initial state to a particular final state is $\int e^{iS(γ)}$, where $γ$ ranges over every possible intermediate field configuration, and $S(γ)$, the action, is a real-valued function of the field configuration. Complex numbers are used to represent the field configurations, but the action is always real, so there's no way for them to "leak through" from the fields to the path integral. It's only in the path integral that quantum amplitudes appear; there are no field quanta, no entangled states, etc., at the level of the action.

In Hamiltonian/canonical quantum mechanics (which is the only kind taught to undergraduates), the complex spinor components are quantum amplitudes. But this can't be essential because canonical QM is in some sense equivalent to path-integral QM. Complex numbers tend to be "genetically dominant" in that if you have real objects $X$ and $Y$, and superscript $C$ means complexification, you tend to have $X^C\otimes Y \cong X\otimes Y^C \cong X^C\otimes Y^C \cong (X\otimes Y)^C$, so the source(s) of the complexification are obscured. I think that's what happens here.

Beyond that, I don't really have any insight about the meaning of complex amplitudes, but you could have a look at Christoph's answer and Ron Maimon's answer to another question about formulating quantum mechanics without complex numbers.

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