{"id":13963,"date":"2024-06-13T15:48:07","date_gmt":"2024-06-13T19:48:07","guid":{"rendered":"https:\/\/www.math.columbia.edu\/~woit\/wordpress\/?p=13963"},"modified":"2024-06-19T10:24:45","modified_gmt":"2024-06-19T14:24:45","slug":"the-mystery-of-spin-2","status":"publish","type":"post","link":"https:\/\/www.math.columbia.edu\/~woit\/wordpress\/?p=13963","title":{"rendered":"The Mystery of Spin"},"content":{"rendered":"<p><em>The following makes no claims to originality or any physical significance on its own.  For a better explanation of some of the math and the physical significance of the use of quaternions here, see <a href=\"https:\/\/johncarlosbaez.wordpress.com\/2022\/11\/30\/this-weeks-finds-lecture-10\/\">this lecture by John Baez<\/a>. <\/p>\n<p>I&#8217;ve been spending a lot of time thinking about spinors and vectors in four dimensions, where I do think there is some important physical significance to the kind of issue discussed here. See chapter 10 <a href=\"https:\/\/www.math.columbia.edu\/~woit\/QFT\/qftmath.pdf\">here<\/a> for something about four dimensions.  A project for the rest of the semester is to write a lot more about this four-dimensional story.<\/em><\/p>\n<p>Until recently I was very fond of the following argument: in three dimensions the relation between spinors and vectors is very simple, with spinors the more fundamental objects.  If one uses the double cover $SU(2)=Spin(3)$ of the rotation group $SO(3)$, the spinor (S) and vector (V) representations satisfy<br \/>\n$$ S\\otimes S = \\mathbf 1 \\oplus V$$<br \/>\nwhich is just the fact well-known to physicists that if you take the tensor product of two spinor representations, you get a scalar and a vector.  The spinors are more fundamental, since you can construct $V$ using $S$, but not the other way around.<\/p>\n<p>I still think spinor geometry is more fundamental than geometry based on vectors.  But it&#8217;s become increasingly clear to me that there is something quite subtle going on here.  The spinor representation is on $S=\\mathbf C^2$, but one wants the vector representation to be on $V_{\\mathbf R}=\\mathbf R^3$, not on its complexification $V=\\mathbf C^3$, which is what one gets by taking the tensor product of spinors.<\/p>\n<p>To get a $V_{\\mathbf R}$ from $V$, one needs an extra piece of structure: a real conjugation on $V$.  This is a map<br \/>\n$$\\sigma:V\\rightarrow V$$<br \/>\nwhich <\/p>\n<ul>\n<li>commutes with the $SU(2)$ action<\/li>\n<li>is antilinear<br \/>\n$$ \\sigma(\\lambda v)=\\overline{\\lambda}\\sigma(v)$$\n<\/li>\n<li>satisfies $\\sigma^2=\\mathbf 1$<\/li>\n<\/ul>\n<p>$V_{\\mathbf R}$ is then the conjugation-invariant subset of $V$.<\/p>\n<p>If we were interested not in usual 3d Euclidean geometry and $Spin(3)$, but in the geometry of $\\mathbf R^3$ with an inner product of $(2,1)$ signature, then the rotation group would be the time-orientation preserving subgroup $SO^+(2,1)\\subset SO(2,1)$, with double cover $SL(2,\\mathbf R)$.  In this case the usual complex conjugations on $\\mathbf C^2$ and $\\mathbf C^3$ provide real conjugation maps that pick out real spinor ($S_{\\mathbf R}=\\mathbf R^2\\subset S$) and vector<br \/>\n$(V_{\\mathbf R}=\\mathbf R^3\\subset V=S\\otimes S)$ representations.<\/p>\n<p>For the case of Euclidean geometry and $Spin(3)$, there is no possible real conjugation map $\\sigma$ on $S$, and while there is a real conjugation map on $V$, it is not complex conjugation.  To better understand what is going on, one can introduce the quaternions $\\mathbf H$, and understand the spin representation in terms of them. The spin group $Spin(3)=SU(2)$ is the group $Sp(1)$ of unit-length quaternions and the spin representation on $S=\\mathbf H$ is just the action on $s\\in S$ of a unit quaternion $q$ by left multiplication<br \/>\n$$s\\rightarrow qs$$<br \/>\n(we could instead define things using right multiplication).<\/p>\n<p>There is an action of $\\mathbf H$ on $S$ commuting with the spin representation, the right action on $S$ by elements $x\\in \\mathbf H$ according to<br \/>\n$$s\\rightarrow s\\overline{q}$$<br \/>\n(this is a right action since $\\overline {q_1q_2}=\\overline q_2\\ \\overline q_1$). <\/p>\n<p>This quaternionic version of the spin representation is a complex representation of the spin group, since the right action by the quaternion $\\mathbf i$ provides a complex structure on $S=\\mathbf H$.  While there are no real conjugation maps $\\sigma$ on the spin representation $S$, there is instead a quaternionic conjugation map, meaning an anti-linear map $\\tau$ commuting with the spin representation and satisfying $\\tau^2=-\\mathbf 1$.  An example is given by right multiplication by $\\mathbf j$<br \/>\n$$\\tau (q)=q\\mathbf j$$<br \/>\nNote that in the above we could have replaced $\\mathbf i$ by any unit-length purely imaginary quaternion and $\\mathbf j$ by any other unit-length purely imaginary quaternion anticommuting with the first.<\/p>\n<p>In general, a representation of a group $G$ on a complex vector space $V$ is called<\/p>\n<ul>\n<li>A real representation if there is a real conjugation $\\sigma$. In this case the group acts on the $\\sigma$-invariant subspace $V_\\mathbf R\\subset V$ and $V$ is the complexification of $V_\\mathbf R$.<\/li>\n<li>A quaternionic representation if there is a quaternionic conjugation $\\tau$.  In this case $\\tau$ makes $V$ a quaternionic vector space, in a way that commutes with the group action.<\/li>\n<\/ul>\n<p>Returning to our original situation of the relation $S\\otimes S= 1 \\oplus V$ between complex representations, $S$ is a quaternionic representation, with a quaternionic conjugation $\\tau$.  Applying $\\tau$ to both terms of the tensor product the minus signs cancel and one gets a real conjugation $\\sigma$ on $V$.<\/p>\n<p>What&#8217;s a bit mysterious is not the above, but the fact that when we do quantum mechanics, we have to work with complex numbers, not quaternions.  We then have to find a consistent way to replace quaternions by complex two by two matrices when they are rotations and and complex column vectors when they are spinors (so $S=\\mathbf C^2$ rather than $\\mathbf H$).<\/p>\n<p>In my <a href=\"https:\/\/www.math.columbia.edu\/%7Ewoit\/QM\/qmbook.pdf\">book on QM and representation theory<\/a> I use a standard sort of choice that identifies $\\mathbf i,\\mathbf j,\\mathbf k$ with corresponding Pauli matrices (up to a factor of $i$):<br \/>\n$$1\\leftrightarrow \\mathbf 1=\\begin{pmatrix}1&#038;0\\\\0&#038;1\\end{pmatrix},\\ \\ \\mathbf i\\leftrightarrow -i\\sigma_1=\\begin{pmatrix}0&#038;-i\\\\ -i&#038;0\\end{pmatrix},\\ \\ \\mathbf j\\leftrightarrow -i\\sigma_2=\\begin{pmatrix}0&#038;-1\\\\ 1&#038;0\\end{pmatrix}$$<br \/>\n$$\\mathbf k\\leftrightarrow -i\\sigma_3=\\begin{pmatrix}-i&#038;0\\\\ 0&#038;i\\end{pmatrix}$$<br \/>\nor equivalently identifies<br \/>\n$$q=q_0 +q_1\\mathbf i +q_2\\mathbf j + q_3\\mathbf k \\leftrightarrow \\begin{pmatrix}q_0-iq_3&#038;-q_2-iq_1\\\\q_2-iq_1 &#038;q_0 +iq_3\\end{pmatrix}$$<\/p>\n<p>Note that this particular choice incorporates the physicist&#8217;s traditional convention distinguishing the $3$-direction as the one for which the spin matrix is diagonalized.<\/p>\n<p>The subtle problem here is the same one discussed above. Just as the vector representation is complex with a non-obvious real conjugation, here complex matrices give not $\\mathbf H$ but its complexification<br \/>\n$$M(2,\\mathbf C)=\\mathbf H\\otimes_{\\mathbf R}\\mathbf C$$<br \/>\n<em>Note added:  complexified quaternions are often called &#8220;biquaternions&#8221;<\/em><br \/>\nThe real conjugation is not complex conjugation, but the non-obvious map<br \/>\n$$\\sigma (\\begin{pmatrix}\\alpha&#038;\\beta \\\\ \\gamma &#038; \\delta\\end{pmatrix})= \\begin{pmatrix}\\overline\\delta &#038;-\\overline\\gamma \\\\ -\\overline\\beta &#038; \\overline\\alpha \\end{pmatrix}$$<\/p>\n<p>Among mathematicians (see for example Keith Conrad&#8217;s <a href=\"https:\/\/kconrad.math.uconn.edu\/blurbs\/ringtheory\/quaternionalg.pdf\">Quaternion Algebras<\/a>), a standard way to consistently identify $\\mathbf H$ with a subset of complex matrices as well as with $\\mathbf C^2$, (giving the spinor representation) is the following:<\/p>\n<ul>\n<li>Identify $\\mathbf C\\subset \\mathbf H$ as<br \/>\n$$z=x+iy\\in \\mathbf C \\leftrightarrow x+\\mathbf i y \\in \\mathbf H$$<\/li>\n<li>Identify $\\mathbf H$ as a complex vector space with $\\mathbf C^2$ by<br \/>\n$$q=z +\\mathbf j w \\leftrightarrow \\begin{pmatrix}z\\\\ w\\end{pmatrix}$$<br \/>\nNote that one needs to be careful about the order of multiplication when writing quaternions this way (where multiplication by a complex number is on the right), since<br \/>\n$$z+w\\mathbf j= z+\\mathbf j\\overline w$$<\/li>\n<li>Identify $\\mathbf H$ as a subset of $M(2,\\mathbf C)$ by<br \/>\n$$q=z +\\mathbf jw \\leftrightarrow \\begin{pmatrix}z&#038;-\\overline{w}\\\\ w&#038; \\overline z\\end{pmatrix}$$<\/li>\n<p>This is determined by requiring that multiplication of quaternions in the spinor story correspond correctly to multiplication of an element of $\\mathbf C^2$ by a matrix.<\/p>\n<p>With this identification<br \/>\n$$\\mathbf i\\leftrightarrow \\begin{pmatrix}i&#038;0\\\\ 0&#038;-i\\end{pmatrix},\\ \\ \\mathbf j\\leftrightarrow \\begin{pmatrix}0&#038;-1\\\\ 1&#038;0\\end{pmatrix},\\ \\ \\mathbf k\\leftrightarrow \\begin{pmatrix} 0&#038;-i\\\\ -i&#038;0\\end{pmatrix}$$<\/p>\n<p>This is a bit different than the Pauli matrix version above, but shares the same real conjugation map identifying $\\mathbf H$ as a subset of $M(2,\\mathbf C)$.<\/ul>\n<p><strong>Update<\/strong>: There&#8217;s a very new video <a href=\"https:\/\/www.youtube.com\/watch?v=IxuJnE_i4nw\">here<\/a>, where Keith Conrad discusses quaternions, especially the case of quaternion algebras over $\\mathbf Q$ and their relation to quadratic reciprocity.<\/p>\n","protected":false},"excerpt":{"rendered":"<p>The following makes no claims to originality or any physical significance on its own. For a better explanation of some of the math and the physical significance of the use of quaternions here, see this lecture by John Baez. I&#8217;ve &hellip; <a href=\"https:\/\/www.math.columbia.edu\/~woit\/wordpress\/?p=13963\">Continue reading <span class=\"meta-nav\">&rarr;<\/span><\/a><\/p>\n","protected":false},"author":2,"featured_media":0,"comment_status":"open","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"_jetpack_newsletter_access":"","_jetpack_dont_email_post_to_subs":false,"_jetpack_newsletter_tier_id":0,"_jetpack_memberships_contains_paywalled_content":false,"_jetpack_feature_clip_id":0,"_jetpack_memberships_contains_paid_content":false,"footnotes":"","jetpack_post_was_ever_published":false},"categories":[1],"tags":[],"class_list":["post-13963","post","type-post","status-publish","format-standard","hentry","category-uncategorized"],"jetpack_featured_media_url":"","jetpack_sharing_enabled":true,"_links":{"self":[{"href":"https:\/\/www.math.columbia.edu\/~woit\/wordpress\/index.php?rest_route=\/wp\/v2\/posts\/13963","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/www.math.columbia.edu\/~woit\/wordpress\/index.php?rest_route=\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/www.math.columbia.edu\/~woit\/wordpress\/index.php?rest_route=\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/www.math.columbia.edu\/~woit\/wordpress\/index.php?rest_route=\/wp\/v2\/users\/2"}],"replies":[{"embeddable":true,"href":"https:\/\/www.math.columbia.edu\/~woit\/wordpress\/index.php?rest_route=%2Fwp%2Fv2%2Fcomments&post=13963"}],"version-history":[{"count":37,"href":"https:\/\/www.math.columbia.edu\/~woit\/wordpress\/index.php?rest_route=\/wp\/v2\/posts\/13963\/revisions"}],"predecessor-version":[{"id":14004,"href":"https:\/\/www.math.columbia.edu\/~woit\/wordpress\/index.php?rest_route=\/wp\/v2\/posts\/13963\/revisions\/14004"}],"wp:attachment":[{"href":"https:\/\/www.math.columbia.edu\/~woit\/wordpress\/index.php?rest_route=%2Fwp%2Fv2%2Fmedia&parent=13963"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/www.math.columbia.edu\/~woit\/wordpress\/index.php?rest_route=%2Fwp%2Fv2%2Fcategories&post=13963"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/www.math.columbia.edu\/~woit\/wordpress\/index.php?rest_route=%2Fwp%2Fv2%2Ftags&post=13963"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}