It’s been taking me forever to sort out and write down the details of implications of the proposal described here. While waiting for that to be done, I thought it might be a good idea to write up one piece of this, which might be some sort of introductory part of the long document I’ve been working on. This at least starts out very simply, explaining what is going on in terms that should be understandable by anyone who has studied the quantization of a spinor field.
I’m not saying anything here about how to use this to get a better unified theory, but am pointing to the precise place in the standard QFT story (the Wick rotation of a Weyl degree of freedom) where I see an opportunity to do something different. This is a rather technical business, which I’d love to convince people is worth paying attention to. Comments from anyone who has thought about this before extremely welcome.
Matter degrees of freedom in the Standard Model are described by chiral spinor fields. Before coupling to gauge fields and the Higgs, these all satisfy the Weyl equation
$$(\frac{\partial}{\partial t}+\boldsymbol\sigma\cdot\boldsymbol\nabla)\psi (t,\mathbf x)=0$$
The Fourier transform of this equation is
$$ (E-\boldsymbol \sigma\cdot \mathbf p)\widetilde{\psi}(E,\mathbf p)=0$$
Multiplying by $(E+\boldsymbol \sigma\cdot \mathbf p)$, solutions satisfy
$$(E^2-|\mathbf p|^2)=0$$
so are supported on the positive and negative light-cones $E=\pm |\mathbf p|$.
The helicity operator
$$\frac{1}{2}\frac{\boldsymbol\sigma\cdot \mathbf p}{|\mathbf p|}$$
will act by $+\frac{1}{2}$ on positive energy solutions, which are said to have “right-handed” helicity. For negative energy solutions, the eigenvalue will be $-\frac{1}{2}$ and these are said to have “left-handed helicity”.
The quantized field $\widehat{\psi}$ will annihilate right-handed particles and create left-handed anti-particles, while its adjoint $\widehat{\psi}^\dagger$ will create right-handed particles and annihilate left-handed anti-particles. One can describe all the Standard model matter particles using such a field. Particles like the electron which have both right-handed and left-handed components can be described by two such chiral fields (note that one is free to interchange what one calls a “particle” or “anti-particle”, or equivalently, which field is $\widehat{\psi}$ and which is the adjoint). Couplings to gauge fields are introduced by changing derivatives to covariant derivatives.
The Lagrangian will be
\begin{equation}
\label{eq:minkowski-lagrangian}
L=\psi^\dagger(\frac{\partial}{\partial t}+\boldsymbol\sigma\cdot\boldsymbol\nabla)\psi
\end{equation}
which is invariant under an action of the group $SL(2,\mathbf C)$, the spin double-cover of the time-orientation preserving Lorentz transformations. To see how this works, note that one can identify Minkowski space-time vectors with two dimensional self-adjoint complex matrices, as in
$$(E,\mathbf p)\leftrightarrow M(E,\mathbf p)=E-\boldsymbol \sigma\cdot \mathbf p=\begin{pmatrix} E-p_3& -p_1+ip_2\\-p_1-ip_2&E+p_3\end{pmatrix}$$
with the Minkowski norm-squared $-E^2-|\mathbf p|^2=-\det M$.
Elements $S\in SL(2,\mathbf C)$ act by
$$M\rightarrow SMS^\dagger$$
which, since it preserves self-adjointness and the determinant, is a Lorentz transformation.
The propagator of a free chiral spinor field in Minkowski space-time is (like other qfts) ill-defined as a function. It is a distribution, generally defined as a certain limit ($i\epsilon$ prescription). This can be done by taking the time and energy variables to be complex, with the propagator a function holomorphic in these variables in certain regions, giving the real time distribution as a boundary value of the holomorphic function. One can instead “Wick rotate” to imaginary time, where the analytically continued propagator becomes a well-defined function.
There is a well-developed formalism for working with Wick-rotated scalar fields in imaginary time, but Wick-rotation of a chiral spinor field is highly problematic. The source of the problem is that in Euclidean signature spacetime, the identification of vectors with complex matrices works differently. Taking the energy to be complex (so of the form $E+is$), Wick rotation gives matrices
$$\begin{pmatrix} is-p_3& -p_1+ip_2\\-p_1-ip_2&is+p_3\end{pmatrix}$$
which are no longer self-adjoint. The determinant of such a matrix is minus the Euclidean norm-squared $(s^2 +|\mathbf p|^2)$. Identifying $\mathbf R^4$ with matrices in this way, the spin double cover of the orthogonal group $SO(4)$ is
$$Spin(4)=SU(2)_L\times SU(2)_R$$
with elements pairs $S_L,S_R$ of $SU(2)$ group elements, acting by
$$\begin{pmatrix} is-p_3& -p_1+ip_2\\-p_1-ip_2&is+p_3\end{pmatrix}\rightarrow S_L\begin{pmatrix} is-p_3& -p_1+ip_2\\-p_1-ip_2&is+p_3\end{pmatrix}S_R^{-1}$$
The Wick rotation of the Minkowski spacetime Lagrangian above will only be invariant under the subgroup $SU(2)\subset SL(2,C)$ of matrices such that $S^\dagger=S^{-1}$ (these are the Lorentz transformations that leave the time direction invariant, so are just spatial rotations). It will also not be invariant under the full $Spin(4)$ group, but only under the diagonal $SU(2)$ subgroup. The conventional interpretation is that a Wick-rotated spinor field theory must contain two different chiral spinor fields, one transforming undert $SU(2)_L$, the other under $SU(2)_R$.
The argument of this preprint is that it’s possible there’s nothing wrong with the naive Wick rotation of the chiral spinor Lagrangian. This makes perfectly good sense, but only the diagonal $SU(2)$ subgroup of $Spin(4)$ acts non-trivially on Wick-rotated spacetime. The rest of the $Spin(4)$ group acts trivially on Wick-rotated spacetime and behaves like an internal symmetry, opening up new possibilities for the unification of internal and spacetime symmetries.
From this point of view, the relation between spacetime vectors and spinors is not the usual one, in a way that doesn’t matter in Minkowski spacetime, but does in Euclidean spacetime. More specifically, in complex spacetime the Spin group is
$$Spin(4,\mathbf C)=SL(2,\mathbf C)_L\times SL(2,\mathbf C)_R$$
there are two kinds of spinors ($S_L$ and $S_R$) and the usual story is that vectors are the tensor product $S_L\otimes S_R$. Restricting to Euclidean spacetime all that happens is that the $SL(2,\mathbf C)$ groups restrict to $SU(2)$.
Something much more subtle though is going on when one restricts to Minkowski spacetime. There the usual story is that vectors are the subspace of $S_L\otimes S_R$ invariant under the action of simultaneously swapping factors and conjugating. These are acted on by the restriction of $Spin(4,\mathbf C)$ to the $SL(2,\mathbf C)$ anti-diagonal subgroup of pairs $(\Omega,\overline{\Omega})$.
The proposal here is that one should instead take complex spacetime vectors to be the tensor product $S_R\otimes \overline{S_R}$, only using right-handed spinors, and the restriction to the Lorentz subgroup to be just the restriction to the $SL(2,\mathbf C)_R$ factor. This is indistinguishable from the usual story if you just think about Minkowski spacetime, since then all you have is one $SL(2,\mathbf C)$, its spin representation $S$ and the conjugate $\overline S$ of this representation. Exactly because of this indistinguishability, one is not changing the symmetries of Minkowski spacetime in any way, in particular not introducing a distinguished time direction.
When one goes to Euclidean spacetime however, things are quite different than the usual story. Now only the $SU(2)_R$ subgroup of $Spin(4)=SU(2)_L\times SU(2)_R$ acts non-trivially on vectors, the $SU(2)_L$ becomes an internal symmtry. Since $S_R$ and $\overline{S_R}$ are equivalent representations, the vector representation is equivalent to $S_R\otimes S_R$ which decomposes into the direct sum of a one-dimensional representation and a three-dimensional representation. Unlike in Minkowski spacetime there is a distinguished direction, the direction of imaginary time.
Having such a distinguished direction is usually considered to be fatal inconsistency. It would be in Minkowski spacetime, but the way quantization in Euclidean quantum field theory works, it’s not an inconsistency. To recover the physical real time, Lorentz invariant theory, one need to pick a distinguished direction and use it (“Osterwalder-Schrader reflection”) to construct the physical state space. Besides the preprint here, see chapter 10 of these notes for a more detailed explanation of the usual story of the different real forms of complexified four-dimensional space.
Peter,
“which be invariant” is probably a typo.
Matt
”The quantum field theory of a free chiral spinor field in Minkowski space-time is (like other qfts) ill-defined.”
???
The free field is is well-defined for any field in a unitary representation of the Poincare group with nonnegative mass and discrete spin. A construction, which is essentially rigorous, is given in Weinberg’s QFT book, Vol. 1. For a fully rigorous construction, see, e.g., Chapter 10 of the book by
M. Talagrand, What Is a Quantum Field Theory? Cambridge Univ. Press, 2022.
Ill-defined terms appear only when one tries to add interactions between quantum fields in 4D spacetime.
Matt Grayson,
Thanks. Fixed.
Arnold Neumaier,
I’ve edited the text to make it less inflammatory.
The main point being made here is that if you think about analytically continuing fields or their expectation values in time (which is basic to axiomatic QFT in the Wightman tradition), something very interesting happens when you try and do this with Weyl spinors. If you think analytic continuation is an unphysical trick you can of course ignore this, but I think you’re missing a fundamental part of the structure of QFT.
Peter,
My point is that you should be mathematically accurate and make no false or misleading claims. Even the new formulation,
”The propagator of a free chiral spinor field in Minkowski space-time is (like other qfts) ill-defined as a function. It is a distribution, generally defined as a certain limit”
is misleading, since the propagator of a free chiral spinor field in Euclidean space-time is still ill-defined as a function, and your statement
”One can instead “Wick rotate” to imaginary time, where the analytically continued propagator becomes a well-defined function.”
is wrong. For the Wick rotation doesn’t eliminate the singular structure – no matter which mass or spin the field has -, it only concentrates it on the diagonal rather than having it on pairs whose difference is null.
Thus whether or not one Wick rotates, one has to deal with distributional propagators and distribution-valued fields.
Arnold Neumaier,
Yes, Schwinger functions are distributional at coinciding points. Informal documents like this blog posting should not be mistaken for precise rigorous mathematics.
The nature of Schwinger functions is however quite different than that of Wightman functions.
But, I’m trying to explain something much more interesting than the usual well-known story about scalar fields, enough about them, this is about spinors.
Probably I have misunderstood your proposal, but wouldn’t correlation functions be non holomorphic, as involving a complex conjugation, every time a 4-vector, e.g. a momentum, appears?
In some of the earlier versions of this proposal I recall you had a story about how the rest of the standard model gauge group + the Higgs showed up naturally in this formalism. Is this still an angle you’re pursuing?
TwoBs,
In the Wick rotation I”m looking at, there’s a distinguished direction in Euclidean spacetime, the imaginary time direction, and one only takes complex coordinates in that direction. In the complex time variable, there will be the usual analyticity coming from the energy being positive.
Thinking of spinor bundles on complexified spacetime as in the twistor picture, the usual story is that the tangent bundle is a holomorphic bundle since it is a tensor product of two different holomorphic spinor bundles. Here the tangent bundle is a product of a holomorphic spinor bundle and its conjugate, anti-holomorphic bundle. I’m still kind of confused about what the right way to think about this if one wants to exploit analyticity.
Mason Kamb,
Yes, it is the speculation about “Euclidean twistor unification” that motivates this. The most unconventional part of that is the claim that one can consistently interpret one of the Euclidean spacetime SU(2)s as an internal symmetry. What I’m trying to work out here is the details of how that happens.
Dear Peter,
“…I’m still kind of confused about what the right way to think about this if one wants to exploit analyticity.”
From your reply, I must say I share the confusion.
It seems to me that, by going to complex 4-momentum by tensoring a right-handed spinor with a complex-conjugate right-handed spinor, you are actually still realizing just $SL(2,C)$ on what you call $M$, as opposed to its complexification $SL(2,C)\times SL(2,C)$ where two independent $SL(2,C)$ matrices would act on the left and on the right of $M$.
That is, you are basically splitting a complex momentum into real and imaginary parts, $p=k+i q$, where both $k$ and $q$ transform under the usual real Lorentz only, as this splitting is preserved by real Lorentz. This is indeed very different than having complex momenta transforming under complexified Lorentz, which is what one typically means by complex momentum. It seems hard to me to reconcile this with nice results such a CPT that requires complexification of $SL(2,C)$ to simply connect to minus the identity.
But most likely I have deeply misunderstood what you are proposing, or things work in unexpected ways?
TwoBs,
Yes, this is quite different than the usual story about complex space-time. The complexification of both Minkowski and Euclidean space just has one SL(2,C) acting on it. Or one could say that both SL(2,C)s still act, but one of them acts trivially (so, is an internal, not space-time symmetry). Yes, the SL(2,C) acts on complexified Minkowsi space as usual on two by two matrices, preserving the decomposition into self-adjoint and skew-adjoint real four-dimensional spaces. On Euclidean space-time, the distinguished imaginary time direction breaks the SL(2,C) down to an SU(2) subgroup.
The complexified tangent bundle is no longer holomorphic, and one no longer has the holomorphic action of SL(2,C) x SL(2,C) that is usually used to analytically continue between Euclidean and Minkowski through four-dimensional complex spacetime. So, one presumably loses the usual CPT theorem. The analytic continuation of Wick rotation is just happening in one variable, not all four.
This is very different than the usual setup. But as far I can see, I can write down the propagator for a Weyl spinor in a consistent way that allows Wick rotation between Euclidean and Minkowski, something one can’t do with the usual analytic continuation. Unclear why I need to get CPT from the usual axiomatic QFT argument.
Again, I’m pretty sure that there’s a better way of understanding what’s going on here than the way I’m looking at things, still thinking about this.
Shouldn’t you do the opposite direction? I.e. starting with one chiral spinor field transforming under Spin(4) in Euclidean space and show that an internal SU(2) pops out in Minkowski space after Wick rotation? Because the latter is what appears to be realized in nature.
anonymous,
The problem is that in the usual formalism, there is no Euclidean theory of a single chiral spinor field. By changing the way Spin(4) acts on vectors, you can have such a thing, with one SU(2) acting trivially (also have such an SU(2) after Wick rotation to Minkowski space).
So far this is just a theory of a single right-handed weyl spinor, not coupled to the internal SU(2). What one wants is a theory of multiple weyl spinors, some coupled to the internal SU(2) as in the SM, some not. The “Euclidean twistor unification” stuff I’ve written is about how to get such symmetries to work using twistor geometry. Still work in progress to understand and write down a detailed theory implementing this.