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 [https://en.wikipedia.org/wiki/Roger_Penrose Sir Roger Penrose] is arguably the most important living descendant of [https://en.wikipedia.org/wiki/Albert_Einstein Albert Einstein's] school of geometric physics. In this episode of [[The Portal Podcast|The Portal]], we avoid the usual questions put to Roger about quantum foundations and quantum consciousness. Instead we go back to ask about the current status of his thinking on what would have been called “Unified Field Theory” before it fell out of fashion a couple of generations ago. In particular, Roger is the dean of one of the only rival schools of thought to have survived the “String Theory wars” of the 1980s-2000s. We discuss his view of this [https://en.wikipedia.org/wiki/Twistor_theory Twistor Theory] and its prospects for unification. Instead of spoon feeding the audience, however, the material is presented as it might occur between colleagues in neighboring fields so that the Portal audience might glimpse something closer to scientific communication rather than made for TV performance pedagogy. We hope you enjoy our conversation with Professor Penrose.
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 }}
 === Spinors ===
-Spinors have two main instantiations: the infinitesimal quantity usually in finite dimensions as the value of a vector field at a point, or as the vector field taken over a finite region of space(time).
+Spinors have two main instantiations: the infinitesimal quantity usually in finite dimensions as the value of a vector field at a point, or as the vector field taken over a finite region of space(time). References will be given after the brief explanations.
+</div>
+; Infinitesimally
+For the infinitesimal quantity in finite dimensions, spinors are constructed via an algebraic means known as representation theory. The key fact that supports the construction is that the special orthogonal Lie groups of rotations <math> SO(n,\mathbb{R}) </math> acting on linear n-dimensional space admit double coverings <math> Spin(n) </math> such that two elements of the spin group correspond to a single rotation. Five algebraic structures comprise the story here: an n-dimensional real vector space, the n-choose-2 dimensional rotation group acting on it, the spin group, another vector space acted upon (the representation) by the spin group to be constructed, and the Clifford algebra of the first real vector space. To distinguish between the first vector space and the second, the elements of the former are simply referred to as vectors and the latter as spinors. This may be confusing because mathematically both are vector spaces whose elements are vectors, however physically the vectors of the first space have a more basic meaning as directions in physical coordinate space.
+</div>
+:;1)
+::The Clifford algebra is constructed directly from the first vector space, by formally/symbolically multiplying vectors and simplifying these expressions according to the rule: <math> v\cdot v = -q(v)1 </math> where q is the length/quadratic form of the vector that the rotations must preserve, or utilizing the polarization identities of quadratic forms: <math> 2q(v,w)=q(v+w)-q(v)-q(w)\rightarrow v\cdot w + w\cdot v = -2q(v,w) </math>. The two-input bilinear form <math> q(\cdot ,\cdot ) </math> is (often) concretely the dot product, and the original quadratic form is recovered by plugging in the same vector twice <math> q(v,v) </math>. The intermediate mathematical structure which keeps track of the formal products of the vectors is known as the tensor algebra <math> T(V)</math> and notably does not depend on <math> q </math>. It helps to interpret the meaning of the unit <math> 1 </math> in the relation as another formally adjoined symbol. Then, the resulting Clifford algebra with the given modified multiplication rule/relation depending on <math> q </math> is denoted by <math> \mathcal{Cl}(V,q)=T(V)/(v\cdot v-q(v)1) </math> as the quotient of <math> T(V) </math> by the subspace of expressions which we want to evaluate to 0. This curtails the dimension of the Clifford algebra to <math> 2^n </math> from infinite dimensions.
+</div>
+:;2)
+::The Spin group is then found within the Clifford algebra, and because the fiber of the double cover map <math> Spin(n)\rightarrow SO(n, \mathbb{R}) </math> is discrete, it is of the same dimension n-choose-2. This will not be constructed here, but only the following operations which distribute over sums in the Clifford algebra are needed to get the Spin group and its homomorphism to the rotation group:
+::;a)
+:::an involution <math> \alpha, \alpha^2=id </math> induced by negating the embedded vectors of the Clifford algebra: <math> \alpha (x \cdot y\cdot z) = (-x)\cdot (-y)\cdot (-z)=(-1)^3x\cdot y\cdot z </math>
+::;b)
+:::the transpose <math> (-)^t </math> which reverses the order of any expression, e.g. <math> (x\cdot y \cdot z)^t = z\cdot y\cdot x </math>
+::;c)
+:::the adjoint action, conjugation, by an invertible element <math> \phi </math> of the Clifford algebra: <math> Ad_{\phi}(x)=\phi\cdot x\cdot \phi^{-1} </math>
+:;3)
+::Spinors require more algebra to construct in general, such as understanding the representations of Clifford algebras as algebras of matrices. In the simplest case, one can choose an orthonormal basis of <math>V:  \{e_1,\cdots,e_n\} </math> and correspond these vectors to n <math> 2^k\times 2^k </math> matrices with <math> k=\lfloor n/2\rfloor </math> such that they obey the same relations as in the Clifford algebra: <math> \gamma^{\mu}\gamma^{\nu}+\gamma^{\nu}\gamma^{\mu}= -2\eta^{\mu\nu}\mathbb{I}_{k\times k} </math> where <math> \mathbb{I}_{k\times k} </math> is the <math> k\times k </math> identity and <math> \eta^{\mu\nu} </math> is the matrix of dot products of the orthonormal basis. The diagonal of <math> \eta^{\mu\nu} </math> can contain negative elements as in the case of the Minkowski norm in 4d spacetime, as opposed to the Euclidean dot product where it also has the form of the identity. Then the spinors are complex vectors in <math> \Delta=\mathbb{C}^{2^k} </math>, however an important counterexample where spinors don't have a purely complex structure is the Majorana representation which is conjectured to be values of neutrino wavefunctions.
+</div>
+:;Summary
+::The following diagram summarizes the relationship with the structures so far:
+[[File:Spinor_construction.png|frameless|center]]
+</div>
+::The hook arrows denote the constructed embeddings, the double arrow gives the spin double cover of the rotations, and the downwards vertical arrows are the representations of the two groups acting on their respective vector spaces. The lack of an arrow between the two vector spaces indicates no direct relationship between vectors and spinors, only between their groups. There is however a bilinear relationship, meaning a map which is quadratic in the components of spinors to obtain components of vectors, justifying not only that spin transformations are square roots of rotations but spinors as square roots of vectors. When n is even, the spinor space will decompose into two vector spaces: <math> \Delta = S^+ \oplus S^- </math>. In a chosen spin-basis these bilinear maps will come from evaluating products of spinors from the respective subspaces under the gamma matrices as quadratic forms.
+</div>
+::In the cases of Euclidean and indefinite signatures of quadratic forms, a fixed orthonormal of <math> V </math> can be identified with the identity rotation such that all other bases/frame are related to it by rotations and thus identified with those rotations. Similarly for the spinors, there are "spin frames" which when choosing one to correspond to the identity, biject with the whole spin group. This theory of group representations on vector spaces and spinors in particular were first realized by mathematicians, in particular [https://www.google.com/books/edition/The_Theory_of_Spinors/f-_DAgAAQBAJ?hl=en&gbpv=1&printsec=frontcover Cartan] for spinors, in the study of abstract symmetries and their realizations on geometric objects. However, the next step was taken by physicists.
+</div>
+;Finitely
+Spinor-valued functions or sections of spinor bundles are the primary objects of particle physics, however they can be realized in non-quantum ways as a model for the electromagnetic field per Penrose's books. When Atiyah stated that the geometrical significance of spinors is not fully understood, it is at this level rather than the well-understood representation algebra level. Analyzing the analogue of the vectors in this scenario, they represent tangents through curves at a particular point in space/on a manifold. The quadratic form then becomes a field of its own, operating on the tangent spaces of each point of the manifold independently but in a smoothly varying way. Arc length of curves can be calculated by integrating the norm of tangent vectors to the curve along it. In general relativity, this is the metric that is a dynamical variable as the solution to Einstein's equations. In order to follow the curvature of the manifold, vector fields are only locally vector valued functions, requiring transitions between regions as in the tribar example previously. Similarly, because the spin and orthogonal groups are so strongly coupled, we can apply the same transitions to spinor-valued functions. The physical reason to do this is that coordinate changes which affect vectors, thus also affect spinor fields in the same way.
+[[File:Penroseblackhole.jpg|thumb|Because the Minkowski inner product can be 0, the tangent vectors for which it is form a cone which Penrose depicts as smoothly varying with spacetime]]
+</div>
+The metric directly gives a way to differentiate vector fields, or finitely comparing the values at different points via parallel translation along geodesics (curves with minimal length given an initial point and velocity). Using this, a derivative operator can be given for spinor fields. It is usually written in coordinates with the gamma matrices:
+<math> Ds(x)=\sum_{\mu=1}^n\gamma_{\mu}\nabla_{e_{\mu}}s(x) </math> where the <math> \nabla_{e_{\mu}} </math> are the metric-given derivatives in the direction of an element of an orthonormal basis vector at x. Their difference from the coordinate partial derivatives helps to quantify the curvature. These orthonormal bases also vary with x, making a field of frames which like before can locally be identified with an <math> O(n,\mathbb{R}) </math>-valued function and globally (if it exists) defines an orientation of the manifold.
+</div>
+Dirac first wrote down the operator in flat space with partial derivatives instead of covariant derivatives, trying to find a first-order operator and an equation:
+</div>
+<math> (iD-m)s(x)=0 </math>
+</div>
+whose solutions are also solutions to the second order Klein-Gordon equation
+</div>
+<math> (D^2+m^2)s(x)=(\Delta+m^2)s(x)=0 </math>
+</div>
+But it was Atiyah who actually named the operator, and utilized its geometric significance. On a curved manifold it does not square to the Laplacian, but differs by the scalar curvature:
+</div>
+<math> D^2s(x)=\Delta s(x)+\frac{1}{4}R s(x) </math> This is known as the Lichnerowicz formula.
+=== Spinor References ===
+Clearly some knowledge of linear algebra and Lie groups assists in understanding the construction and meaning of spinors. With our [https://theportal.wiki/wiki/Read list] in mind, other books may more directly approach the topic. Spinors are implicit/given in specific representations in the quantum mechanics and field theory books.
+Penrose's books being given, the following give introductions to these topics at various levels:
+<div class="flex-container" style="clear: both;">
+{{BookListing
+| cover = Garling_Clifford_Algbras.jpg
+| link = Clifford Algebras: An Introduction (Book)
+| title =
+| desc = Use this book to learn about Clifford algebras and spinors directly, it covers the necessary prerequisite linear algebra and group theory but only briefly touches on the relation to curvature.
+}}
+{{BookListing
+| cover = Fulton-Harris Representation Theory cover.jpg
+| link = Representation Theory (Book)
+| title =
+| desc = If following our main list [https://theportal.wiki/wiki/Read here], you will encounter Clifford algebras and spin representations here.
+}}
+{{BookListing
+| cover = Woit Quantum Theory, Groups and Representations.png
+| link = Quantum Theory, Groups and Representations (Book)
+| title =
+| desc = Less general discussion of spin representations, but with focus on the low dimensional examples in quantum physics.
+}}
+{{BookListing
+| cover = Lawson Spin Geometry cover.jpg
+| link = Spin Geometry (Book)
+| title =
+| desc = Immediately introduces Clifford algebras and spin representations, demanding strong linear algebra. The remainder of the book extensively introduces the theory of the Dirac operator, Atiyah-Singer Index theorem, and some assorted applications in geometry.
+}}
 </div>
-For the infinitesimal quantity in finite dimensions, spinors are constructed via an algebraic means known as representation theory. The key fact that supports the construction is that the Lie groups of rotations <math> SO(n,\mathbb{R}) </math> acting on linear n-dimensional space admit double coverings <math> Spin(n) </math> such that two elements of the spin group correspond to a single rotation.
 == Notes ==