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## Substance

*A summary of Penrose's summary of physics and math.*

William Rowan Hamilton discovered the following in 1843: \[i^2 = j^2 = k^2 = ijk = -1\] (i, j, k are bolded - say my notes, and therefore probs vectors)
where each of \(i\), \(j\), and \(k\) are an independent 'square root of -1' \(\bullet\) a quaternion ... 3 Blue 1 Brown - Visualizing quaternions (4d numbers) with stereographic projection-Quaternions and 3d rotation, explained interactively
\[q = t + ui + vj + wk\] with \(t\), \(u\), \(v\), \(w\) real
\(\bullet\) quaternions are not commutative! (\(ab \neq ba\)) \(\bullet\) properties: ---1. commutative addition \(a + b = b + a\) ---2. associative addition \(a + (b + c) = (a + b) + c\) ---3. associative multiplication \(a(bc) = (ab)c\) ---4. distributive laws \((a(b+c) = ac + ab\) and \((a+b)c = ac + bc\) ---5. additive identity \(a+0 = a\) ---6. multiplicative identity \(|a + a| = a\) The properties numbered 1-5 meet the requirements for this to be an algebraic ring while the 6th property further allows for the classification as a ring with identity. \(\bullet\) quaternions form a 4-dimensional vector space over the real numbers since they have 4 independent basis quantities \l\), \(i\), \(j\), \(k\) spanning the space of quanternion \(\bullet\) due to the multiplication law, they provide an algebra over the real numbers \(\bullet\) they also have the structure of a division ring since they have multiplicative inverses: \[q^{-1}q = qq^{-1} = 1\] \[q^{-1} = \bar{q}(q\bar{q})^{-1}\] where the latter is the explicit definition of the inverse. \(\bullet\) the conjugate is defined as \[\bar{q} = t - ui - vj - wk\] \(\bullet\) \(q\bar{q} = t^{2} + u^{2} + v^{2} + w^{2}\) which is a real, nonzero number unless \(q = 0\) \(\bullet\) quaternions have not proven to be very useful in physics, for example \(q\bar{q}\) does not have the correct signature (I suppose he is referring to the Minkowski signature) for use in relativity \(\bullet\) cannot define quaternion-holomorphic functions because \(\bar{q}\) would have to be such a function of \(q\) and we previously defined holomorphic functions as those of the complex variable \(z\), independent of \(\bar{z}\) ## Geometry of Quaternions
Let \(i\), \(j\), \(k\) be three orthogonal axes in Euclidean space. Then, multiplying \(i\) is a rotation about the \(i\) axis with the \((j, k)\)-plane as the complex place and likewise for \(j\) and \(k\). In order to satisfy the properties of quaternions however, rather than rotating 90\(^{\circ}\) as before, we must now rotate by 180\(^{\circ}\)/\(\pi\) radians. An important property occurs when rotating by \(2\pi\), we find that \(i^{2} = 2\) rather than \(-1\). \(\star\) This yields the notion of a spinor! \(\star\) That is, "an object with turns into its negative when it undergoes a complete rotation through \(2\pi\)" (pg. 204). Thus, the number of \(2\pi\) rotations have a parity based on if they are even or odd. "To picture a spinoral object, we can think of an ordinary object in space, but where there is an imaginary flexible attachment to some fixed external structure..." (pg. 205).
## Clifford Algebras
\(\bullet\) to discuss higher dimensions, must consider what the analogue of a rotation is about an axis is in n dimensions, such a rotation is about an "axis" that is an \(n-2)\) dimensional space.
---- in higher dimensions, a composition of rotations about an \((n-2)\) dimensional axis is not always a rotation about an \((n-2)\) dimensional axis \(\bullet\) more simple to first consider reflections about an \((n-1)\) dimensional hyperplane where a composition of 2 such reflections wrt. 2 hyperplanes constitutes a \(\pi\) rotation (so can consider such reflections as secondary entities) \(\bullet\) let \(\gamma_{1}\). \(\gamma_{2}\), \(\gamma_{3}\), ... , \(\gamma_{n}\) be reflections where \(\gamma_{r}\) reverses the \(r^{th}\) coordinate axis there are \(n\) quaternion-like relations: \(\gamma_{1}^{2} = -1\), \(\gamma_{2}^{2}= -1\), \(\gamma_{3}^{2} = -1\), ..., \(\gamma_{n}^{2} = -1\) then the \(\pi\)-rotation entities are products of \(\gamma_{i}\)'s with the anticommutation property: \(\gamma_{p}\gamma_{q} = -\gamma_{q}\gamma_{p}\) with \(p \neq q\). (So for 3D, can express \(i = \gamma_{2}\gamma_{3}\), \(j = \gamma_{3}\gamma_{1}\), \(k = \gamma_{1}\gamma_{2}\).) \(\bullet\) General element of a Clifford algebra for \(n\) dimensional space is a linear combination of products of distinct \(\gamma\)'s 1st order = the \(n\) \(\gamma_{i}\)'s 2nd order = the \(\frac{1}{2}n(n-1)\) independent products \(\gamma_{p}\gamma_{q}\) (with \(p < q\) 3rd order = the \(\frac{1}{6}n(n-1)(n-2)\) 3 \(\gamma\) products ...and so on The sum of "entities" is thus \[1 + n + \frac{1}{2}n(n-1) + \frac{1}{6}n(n-1)(n-2) + ... + 1 = 2^{n}\] This is a sum of \(n\) choose \(k\) for \(k\) from 0 to \(n\). \(\bullet\) The Clifford algebra is therefore a \(2^{n}\)-dimensional algebra over the reals ---- a ring with identify ---- unlike quaternions though, they are not a division ring "A spinor may be thought of as an object upon which the elements of the Clifford algebra act as operators" (pg. 210). \(\bullet\) the spin space (space of spinors) in \(n\) dimensions: ---- for odd \(n\): \(2^{(n-1)/2}\) dimensional ---- for even \(n\): \(2^{n/2}\) dimensional...this splits into 2 spaces of reduced or half spinors of \(2^{(n-2)/2}\) dimensions each. So each element of the full spinor has a component from each of these subspaces. \(\bullet\) For even \(n\)-dimensional space, a reflection converts one half-spinor into the other. \(\bullet\) Elements of reduced spin space have a chirality (as do those of the opposite space with opposite chirality). ## Grassmann Algebras
\(\bullet\) Have anticommuting elements \(\eta_{1}, \eta_{2}, ..., \eta_{n}\) where each \(\eta_{i}\) squares to 0:
\(\eta_{1}^{2} = 0\), \(\eta_{2}^{2} = 0\), ..., \(\eta_{n}^{2} = 0\) \(\bullet\) Still have \(\eta_{p}\eta_{q} = -\eta_{q}\eta_{p}\) BUT can have \(p = q\) \(\bullet\) Unlike Clifford algebras, does not require there to be a metric on the space \(\bullet\) Can think of each \(\eta_{i}\) as a line element/vector associated to one of the \(n\) coordinate axis (Interesting note: the axes do not need to be orthogonal) \(\bullet\) Let us ahve 2 vectors at the origin: \[a = a_{1}\eta_{1} + a_{2}\eta_{2} + ... + a_{n}\eta_{n}\] \[b = b_{1}\eta_{1} + b_{2}\eta_{2} + ... + b_{n}\eta_{n}\] and denote the Grassmann/wedge product as \(a \wedge b\) \(\bullet\) we can find that \(a\wedge b = -b\wedge a\) \(\bullet\) \(a \wedge b\) = the plane element spanned by the two vectors \(\bullet\) We find the coefficients/components of \(a \wedge b\) to be the quantities \(a[_{p}b_{q}]\) with the square brackets denoting antisymmetrization. Antisymmetrization is defined generally by \[A_{[pq]} = \frac{1}{2}(A_{pq}-A_{qp})\] or more specifically here, \[a[_{p}b_{q}] = \frac{1}{2}(a_{p}b_{q} - a_{q}b_{p})\] \(\bullet\) \(a \wedge b \wedge c\) can then be thought of as a 3D plane element having an orientation and magnitude as before. One finds anticommutation properties: \[a \wedge b \wedge c = b \wedge c \wedge a = c \wedge a \wedge b = -b\wedge a \wedge c = -a\wedge c \wedge b = -c\wedge b \wedge a\] Notice how the odd permutations have a negative sign and the even ones do not. \(\bullet\) Grassmann algebra is a graded algebra ---- contains \(r^{th}\)-order elements with \(r\) the number of \(\eta\)'s that are wedge-producted togehter within the expression ---- the number \(r\) can range from 1 to \(n\) and is called the grade of the element of the Grassmann algebra Note: a general element of the algebra of grade \(r\) with \(r = 3\) for example, can be a sum of expressions of the form \(a \wedge b \wedge c\) For \(P\) an element of grade \(p\) and \(Q\) and element of grade \(q\), their \((p+q)\)-grade wedge product \(P\wedge Q\) has components \(P[_{a...c}Q_{d...f}]\) where \(P_{a..c}\) and \(Q_{d..f}\) are the components of \(P\) and \(Q\), respectively. \[P \wedge Q = \begin{cases}+Q\wedge P & p\textrm{, or } q\textrm{, or both are even}\\ -Q\wedge P & p\textrm{ and }q\textrm{ are both odd} \\ \end{cases}\] \(\star\star\star\) Should not confuse grade with degree of the form though I think they can be the same...idk, unsure at time of typing. "We may also add together elements of different grades to obtain a 'mixed' quantity that does not have any particular grade" (pg. 215). ...To which I wrote WHY?
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