|
1 | | - |
2 | 1 | { |
3 | 2 | "cells" : |
4 | 3 | [ |
5 | | - |
6 | 4 | { |
7 | 5 | "cell_origin" : "client", |
8 | 6 | "cell_type" : "latex", |
9 | 7 | "cells" : |
10 | 8 | [ |
11 | | - |
12 | 9 | { |
13 | 10 | "cell_origin" : "client", |
14 | 11 | "cell_type" : "latex_view", |
|
18 | 15 | "hidden" : true, |
19 | 16 | "source" : "\\property{AntiCommuting}{Make objects anti-commuting.}\n\nMakes components anti-commuting, for example" |
20 | 17 | }, |
21 | | - |
22 | 18 | { |
23 | 19 | "cell_origin" : "client", |
24 | 20 | "cell_type" : "input", |
25 | 21 | "cells" : |
26 | 22 | [ |
27 | | - |
28 | 23 | { |
29 | 24 | "cell_origin" : "server", |
30 | 25 | "cell_type" : "latex_view", |
31 | | - "source" : "\\begin{dmath*}{}\\text{Attached property AntiCommuting to~}(A, B).\\end{dmath*}" |
| 26 | + "source" : "\\begin{dmath*}{}\\text{Attached property AntiCommuting to~}\\left[A,~\\discretionary{}{}{} B\\right].\\end{dmath*}" |
32 | 27 | }, |
33 | | - |
34 | 28 | { |
35 | 29 | "cell_origin" : "server", |
36 | 30 | "cell_type" : "latex_view", |
37 | 31 | "source" : "\\begin{dmath*}{}B A\\end{dmath*}" |
38 | 32 | }, |
39 | | - |
40 | 33 | { |
41 | 34 | "cell_origin" : "server", |
42 | 35 | "cell_type" : "latex_view", |
|
45 | 38 | ], |
46 | 39 | "source" : "{A,B}::AntiCommuting;\nex:=B A;\nsort_product(_);" |
47 | 40 | }, |
48 | | - |
49 | 41 | { |
50 | 42 | "cell_origin" : "client", |
51 | 43 | "cell_type" : "latex", |
52 | 44 | "cells" : |
53 | 45 | [ |
54 | | - |
55 | 46 | { |
56 | 47 | "cell_origin" : "client", |
57 | 48 | "cell_type" : "latex_view", |
|
61 | 52 | "hidden" : true, |
62 | 53 | "source" : "It also works for objects with indices:" |
63 | 54 | }, |
64 | | - |
65 | 55 | { |
66 | 56 | "cell_origin" : "client", |
67 | 57 | "cell_type" : "input", |
68 | 58 | "cells" : |
69 | 59 | [ |
70 | | - |
71 | 60 | { |
72 | 61 | "cell_origin" : "server", |
73 | 62 | "cell_type" : "latex_view", |
74 | 63 | "source" : "\\begin{dmath*}{}\\psi_{m} \\chi \\psi_{n}\\end{dmath*}" |
75 | 64 | }, |
76 | | - |
77 | 65 | { |
78 | 66 | "cell_origin" : "server", |
79 | 67 | "cell_type" : "latex_view", |
|
82 | 70 | ], |
83 | 71 | "source" : "{\\psi_{m}, \\chi}::AntiCommuting.\nex:= \\psi_{m} \\chi \\psi_{n};\nsort_product(_);" |
84 | 72 | }, |
85 | | - |
86 | 73 | { |
87 | 74 | "cell_origin" : "client", |
88 | 75 | "cell_type" : "latex", |
89 | 76 | "cells" : |
90 | 77 | [ |
91 | | - |
92 | 78 | { |
93 | 79 | "cell_origin" : "client", |
94 | 80 | "cell_type" : "latex_view", |
|
98 | 84 | "hidden" : true, |
99 | 85 | "source" : "If you want a pattern like \\verb|\\psi_{m}| to anti-commute with\nitself, you should use the \\prop{SelfAntiCommuting} property instead." |
100 | 86 | }, |
101 | | - |
102 | 87 | { |
103 | 88 | "cell_origin" : "client", |
104 | 89 | "cell_type" : "latex", |
105 | 90 | "cells" : |
106 | 91 | [ |
107 | | - |
108 | 92 | { |
109 | 93 | "cell_origin" : "client", |
110 | 94 | "cell_type" : "latex_view", |
|
114 | 98 | "hidden" : true, |
115 | 99 | "source" : "You can think about the difference\nbetween \\prop{SelfAntiCommuting} and \\prop{AntiCommuting} in\nthe following way. If \\verb|A_{m n}| is \\prop{SelfAntiCommuting}, it\nmeans that for each value of the indices the expression \\verb|A_{m n}|\nis an operator which anti-commutes with the operator for any other\nvalue of the indices. The matrix~$A$ is thus a matrix of\noperator-valued components which mutually anti-commute. On the other\nhand, if \\verb|A| and\n \\verb|B| are declared to\nbe \\prop{AntiCommuting}, then these can be viewed as two matrices of\ncommuting components, whose matrix product satisfies~$A B = - B A$." |
116 | 100 | }, |
117 | | - |
118 | 101 | { |
119 | 102 | "cell_origin" : "client", |
120 | 103 | "cell_type" : "latex", |
121 | 104 | "cells" : |
122 | 105 | [ |
123 | | - |
124 | 106 | { |
125 | 107 | "cell_origin" : "client", |
126 | 108 | "cell_type" : "latex_view", |
127 | | - "source" : "If you attach the \\prop|AntiCommuting| property to an object\nwith an \\prop{ImplicitIndex} property, the commutation property does\nnot refer to the object as a whole, but rather to its components. The\nlogic behind that becomes clear when considering e.g.~spinor bilinears," |
| 109 | + "source" : "If you attach the \\prop{AntiCommuting} property to an object\nwith an \\prop{ImplicitIndex} property, the commutation property does\nnot refer to the object as a whole, but rather to its components. The\nlogic behind that becomes clear when considering e.g.~spinor bilinears," |
128 | 110 | } |
129 | 111 | ], |
130 | 112 | "hidden" : true, |
131 | | - "source" : "If you attach the \\prop|AntiCommuting| property to an object\nwith an \\prop{ImplicitIndex} property, the commutation property does\nnot refer to the object as a whole, but rather to its components. The\nlogic behind that becomes clear when considering e.g.~spinor bilinears," |
| 113 | + "source" : "If you attach the \\prop{AntiCommuting} property to an object\nwith an \\prop{ImplicitIndex} property, the commutation property does\nnot refer to the object as a whole, but rather to its components. The\nlogic behind that becomes clear when considering e.g.~spinor bilinears," |
132 | 114 | }, |
133 | | - |
134 | 115 | { |
135 | 116 | "cell_origin" : "client", |
136 | 117 | "cell_type" : "input", |
137 | 118 | "cells" : |
138 | 119 | [ |
139 | | - |
140 | 120 | { |
141 | 121 | "cell_origin" : "server", |
142 | 122 | "cell_type" : "latex_view", |
143 | | - "source" : "\\begin{dmath*}{}\\text{Attached property Spinor to~}(\\chi, \\psi).\\end{dmath*}" |
| 123 | + "source" : "\\begin{dmath*}{}\\text{Attached property Spinor to~}\\left[\\chi,~\\discretionary{}{}{} \\psi\\right].\\end{dmath*}" |
144 | 124 | }, |
145 | | - |
146 | 125 | { |
147 | 126 | "cell_origin" : "server", |
148 | 127 | "cell_type" : "latex_view", |
149 | | - "source" : "\\begin{dmath*}{}\\text{Attached property AntiCommuting to~}(\\chi, \\psi).\\end{dmath*}" |
| 128 | + "source" : "\\begin{dmath*}{}\\text{Attached property AntiCommuting to~}\\left[\\chi,~\\discretionary{}{}{} \\psi\\right].\\end{dmath*}" |
150 | 129 | }, |
151 | | - |
152 | 130 | { |
153 | 131 | "cell_origin" : "server", |
154 | 132 | "cell_type" : "latex_view", |
155 | 133 | "source" : "\\begin{dmath*}{}\\text{Attached property DiracBar to~}\\bar{\\#}.\\end{dmath*}" |
156 | 134 | }, |
157 | | - |
158 | 135 | { |
159 | 136 | "cell_origin" : "server", |
160 | 137 | "cell_type" : "latex_view", |
161 | | - "source" : "\\begin{dmath*}{}\\text{Attached property GammaMatrix to~}\\Gamma(\\#).\\end{dmath*}" |
| 138 | + "source" : "\\begin{dmath*}{}\\text{Attached property GammaMatrix to~}\\Gamma\\left(\\#\\right).\\end{dmath*}" |
162 | 139 | }, |
163 | | - |
164 | 140 | { |
165 | 141 | "cell_origin" : "server", |
166 | 142 | "cell_type" : "latex_view", |
167 | | - "source" : "\\begin{dmath*}{}\\text{Attached property SortOrder to~}(\\chi, \\psi).\\end{dmath*}" |
| 143 | + "source" : "\\begin{dmath*}{}\\text{Attached property SortOrder to~}\\left[\\chi,~\\discretionary{}{}{} \\psi\\right].\\end{dmath*}" |
168 | 144 | }, |
169 | | - |
170 | 145 | { |
171 | 146 | "cell_origin" : "server", |
172 | 147 | "cell_type" : "latex_view", |
|
175 | 150 | ], |
176 | 151 | "source" : "{\\chi, \\psi}::Spinor(dimension=10, type=MajoranaWeyl);\n{\\chi, \\psi}::AntiCommuting;\n\\bar{#}::DiracBar;\n\\Gamma{#}::GammaMatrix;\n{\\chi, \\psi}::SortOrder;\nex:=\\bar{\\psi} \\Gamma_{m n p} \\chi;" |
177 | 152 | }, |
178 | | - |
179 | 153 | { |
180 | 154 | "cell_origin" : "client", |
181 | 155 | "cell_type" : "input", |
182 | 156 | "cells" : |
183 | 157 | [ |
184 | | - |
185 | 158 | { |
186 | 159 | "cell_origin" : "server", |
187 | 160 | "cell_type" : "latex_view", |
|
190 | 163 | ], |
191 | 164 | "source" : "sort_product(_);" |
192 | 165 | }, |
193 | | - |
194 | 166 | { |
195 | 167 | "cell_origin" : "client", |
196 | 168 | "cell_type" : "input", |
197 | 169 | "cells" : |
198 | 170 | [ |
199 | | - |
200 | 171 | { |
201 | 172 | "cell_origin" : "server", |
202 | | - "cell_type" : "error", |
203 | | - "source" : "{\\color{red}{\\begin{verbatim}Traceback (most recent call last):\n File \"<string>\", line 1, in <module>\nNameError: name 'sort_spinors' is not defined\n\\end{verbatim}}}" |
| 173 | + "cell_type" : "latex_view", |
| 174 | + "source" : "\\begin{dmath*}{}\\bar{\\chi} \\Gamma_{m n p} \\psi\\end{dmath*}" |
204 | 175 | } |
205 | 176 | ], |
206 | 177 | "source" : "sort_spinors(_);" |
207 | 178 | }, |
208 | | - |
209 | 179 | { |
210 | 180 | "cell_origin" : "client", |
211 | 181 | "cell_type" : "latex", |
212 | 182 | "cells" : |
213 | 183 | [ |
214 | | - |
215 | 184 | { |
216 | 185 | "cell_origin" : "client", |
217 | 186 | "cell_type" : "latex_view", |
|
221 | 190 | "hidden" : true, |
222 | 191 | "source" : "Here \\algo{sort_product} did not act because both the spinors and\nthe gamma matrices have the \\prop{ImplicitIndex} property and\nthere are thus no simple rules for their re-ordering. However,\nthe \\algo{sort_spinors} algorithm did act, and took into account\nthe fact that the components of the spinors are anti-commuting." |
223 | 192 | }, |
224 | | - |
225 | 193 | { |
226 | 194 | "cell_origin" : "client", |
227 | 195 | "cell_type" : "input", |
228 | 196 | "source" : "" |
229 | 197 | } |
230 | 198 | ], |
231 | 199 | "description" : "Cadabra JSON notebook format", |
232 | | - "version" : 1.0 |
| 200 | + "version" : 1 |
233 | 201 | } |
234 | | - |
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