The Dirac equation in Robertson-Walker SpacetimeMark D. Roberts (ORCID link)The following program is perhaps the simplest case which has a fermionic spinor field and a non-flat spacetime. The massless m=0 Dirac equation (sometimes called the Weyl equation) is derived in k=0 Robertson-Walker geometry. To do this it is necessary to introduce a spinorial covariant derivative, the spinorial term involves first derivatives of the metric and 'shuffles' up the order of the four components of the wave function column vector, the way that the shuffle happens is dependant on the representation of the Dirac gamma matrices. In the present case there is only one first order derivative of the metric the Hubble parameter LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYnLUkjbWlHRiQ2I1EhRictRiM2KC1GLDYmUSJIRicvJSdpdGFsaWNHUSV0cnVlRicvJStleGVjdXRhYmxlR1EmZmFsc2VGJy8lLG1hdGh2YXJpYW50R1EnaXRhbGljRictSShtZmVuY2VkR0YkNiUtRiM2JS1GLDYmUSJ0RidGNEY3RjpGNy9GO1Enbm9ybWFsRidGN0ZFLUkjbW9HRiQ2LlEiPUYnRjdGRS8lJmZlbmNlR0Y5LyUqc2VwYXJhdG9yR0Y5LyUpc3RyZXRjaHlHRjkvJSpzeW1tZXRyaWNHRjkvJShsYXJnZW9wR0Y5LyUubW92YWJsZWxpbWl0c0dGOS8lJ2FjY2VudEdGOS8lJ2xzcGFjZUdRLDAuMjc3Nzc3OGVtRicvJSdyc3BhY2VHRmVuLUkmbWZyYWNHRiQ2KC1GIzYnLUkmbW92ZXJHRiQ2JS1GIzYlLUYsNiZRImFGJ0Y0RjdGOkY3RkUtRkg2L1EiLkYnRjcvJSxwbGFjZWhvbGRlckdGNkZFL0ZMUSZ1bnNldEYnL0ZORltwL0ZQRltwL0ZSRltwL0ZURltwL0ZWRltwL0ZYRltwL0ZaUSYwLjBlbUYnL0ZnbkZjcC9GWEY2RitGPUY3RkUtRiM2JkZib0Y9RjdGRS8lLmxpbmV0aGlja25lc3NHUSIxRicvJStkZW5vbWFsaWduR1EnY2VudGVyRicvJSludW1hbGlnbkdGXXEvJSliZXZlbGxlZEdGOUY3RkVGK0Y3RkU=. To produce the Dirac equation the covariant derivative is left multiplied by the Dirac gamma matrices giving LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2I1EhRic=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 the wavefunction column vector LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUkjbWlHRiQ2JlEnJiM5MzY7RicvJSdpdGFsaWNHUSZmYWxzZUYnLyUrZXhlY3V0YWJsZUdGMS8lLG1hdGh2YXJpYW50R1Enbm9ybWFsRidGMkY0 in its original order. The program can be genralized to include angular terms m,k not zero and so on, and hopefully at some stage to supergravity. The program takes out components because maple does not yet include Dirac gammas in arbitrary spacetimes or sums over tetrad indices. If/when these features are introduced it should be much reduced. 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load the metric, it is 12,9,2 but is here done 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 metric, tetrad and internal metric 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LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYnLUkjbWlHRiQ2JVElZXRhX0YnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1JKG1mZW5jZWRHRiQ2Ji1GIzYlLUYsNiNRIUYnLyUrZXhlY3V0YWJsZUdRJmZhbHNlRicvRjNRJ25vcm1hbEYnRkAvJSVvcGVuR1EiW0YnLyUmY2xvc2VHUSJdRictSSNtb0dGJDYtUSI7RidGQC8lJmZlbmNlR0Y/LyUqc2VwYXJhdG9yR0YxLyUpc3RyZXRjaHlHRj8vJSpzeW1tZXRyaWNHRj8vJShsYXJnZW9wR0Y/LyUubW92YWJsZWxpbWl0c0dGPy8lJ2FjY2VudEdGPy8lJ2xzcGFjZUdRJjAuMGVtRicvJSdyc3BhY2VHUSwwLjI3Nzc3NzhlbUYnRj1GQA==Now the spacetime indexed Dirac gamma matrices are calculated. This is done explicitly for components.LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYtLUkjbWlHRiQ2JVEjZ3hGJy8lJ2l0YWxpY0dRJXRydWVGJy8lLG1hdGh2YXJpYW50R1EnaXRhbGljRictSSNtb0dGJDYtUSomY29sb25lcTtGJy9GM1Enbm9ybWFsRicvJSZmZW5jZUdRJmZhbHNlRicvJSpzZXBhcmF0b3JHRj0vJSlzdHJldGNoeUdGPS8lKnN5bW1ldHJpY0dGPS8lKGxhcmdlb3BHRj0vJS5tb3ZhYmxlbGltaXRzR0Y9LyUnYWNjZW50R0Y9LyUnbHNwYWNlR1EsMC4yNzc3Nzc4ZW1GJy8lJ3JzcGFjZUdGTC1GLDYlUSdEZ2FtbWFGJ0YvRjItSShtZmVuY2VkR0YkNiYtRiM2KC1GNjYtUSJ8aXJGJ0Y5RjtGPkZARkJGREZGRkgvRktRLDAuMTExMTExMWVtRicvRk5GZW4tSSNtbkdGJDYkUSIxRidGOS1GNjYtUSIsRidGOUY7L0Y/RjFGQEZCRkRGRkZIL0ZLUSYwLjBlbUYnL0ZOUSwwLjMzMzMzMzNlbUYnLUYsNiVRJ21hdHJpeEYnRi9GMi8lK2V4ZWN1dGFibGVHRj1GOUY5LyUlb3BlbkdRIltGJy8lJmNsb3NlR1EiXUYnLUY2Ni1RIi5GJ0Y5RjtGPkZARkJGREZGRkhGX28vRk5GYG8tRjY2LVEifkYnRjlGO0Y+RkBGQkZERkZGSEZfb0ZhcC1GLDYlUSNlX0YnRi9GMi1GUzYmLUYjNihGZ25GW29GV0ZnbkZmb0Y5RjlGaG9GW3AtRjY2LVEiO0YnRjlGO0Zeb0ZARkJGREZGRkhGX29GTUZmb0Y5LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYtLUkjbWlHRiQ2JVEjZ3lGJy8lJ2l0YWxpY0dRJXRydWVGJy8lLG1hdGh2YXJpYW50R1EnaXRhbGljRictSSNtb0dGJDYtUSomY29sb25lcTtGJy9GM1Enbm9ybWFsRicvJSZmZW5jZUdRJmZhbHNlRicvJSpzZXBhcmF0b3JHRj0vJSlzdHJldGNoeUdGPS8lKnN5bW1ldHJpY0dGPS8lKGxhcmdlb3BHRj0vJS5tb3ZhYmxlbGltaXRzR0Y9LyUnYWNjZW50R0Y9LyUnbHNwYWNlR1EsMC4yNzc3Nzc4ZW1GJy8lJ3JzcGFjZUdGTC1GLDYlUSdEZ2FtbWFGJ0YvRjItSShtZmVuY2VkR0YkNiYtRiM2Jy1GNjYtUSJ8aXJGJ0Y5RjtGPkZARkJGREZGRkgvRktRLDAuMTExMTExMWVtRicvRk5GZW4tSSNtbkdGJDYkUSIyRidGOS1GNjYtUSIsRidGOUY7L0Y/RjFGQEZCRkRGRkZIL0ZLUSYwLjBlbUYnL0ZOUSwwLjMzMzMzMzNlbUYnLUYsNiVRJ21hdHJpeEYnRi9GMkY5RjkvJSVvcGVuR1EiW0YnLyUmY2xvc2VHUSJdRictRjY2LVEiLkYnRjlGO0Y+RkBGQkZERkZGSEZfby9GTkZgby1GNjYtUSJ+RidGOUY7Rj5GQEZCRkRGRkZIRl9vRl9wLUYsNiVRI2VfRidGL0YyLUZTNiYtRiM2J0ZnbkZbb0ZXRmduRjlGOUZmb0Zpby1GNjYtUSI7RidGOUY7Rl5vRkBGQkZERkZGSEZfb0ZNLyUrZXhlY3V0YWJsZUdGPUY5LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYtLUkjbWlHRiQ2JVEjZ3pGJy8lJ2l0YWxpY0dRJXRydWVGJy8lLG1hdGh2YXJpYW50R1EnaXRhbGljRictSSNtb0dGJDYtUSomY29sb25lcTtGJy9GM1Enbm9ybWFsRicvJSZmZW5jZUdRJmZhbHNlRicvJSpzZXBhcmF0b3JHRj0vJSlzdHJldGNoeUdGPS8lKnN5bW1ldHJpY0dGPS8lKGxhcmdlb3BHRj0vJS5tb3ZhYmxlbGltaXRzR0Y9LyUnYWNjZW50R0Y9LyUnbHNwYWNlR1EsMC4yNzc3Nzc4ZW1GJy8lJ3JzcGFjZUdGTC1GLDYlUSdEZ2FtbWFGJ0YvRjItSShtZmVuY2VkR0YkNiYtRiM2Jy1GNjYtUSJ8aXJGJ0Y5RjtGPkZARkJGREZGRkgvRktRLDAuMTExMTExMWVtRicvRk5GZW4tSSNtbkdGJDYkUSIzRidGOS1GNjYtUSIsRidGOUY7L0Y/RjFGQEZCRkRGRkZIL0ZLUSYwLjBlbUYnL0ZOUSwwLjMzMzMzMzNlbUYnLUYsNiVRJ21hdHJpeEYnRi9GMkY5RjkvJSVvcGVuR1EiW0YnLyUmY2xvc2VHUSJdRictRjY2LVEiLkYnRjlGO0Y+RkBGQkZERkZGSEZfby9GTkZgby1GNjYtUSJ+RidGOUY7Rj5GQEZCRkRGRkZIRl9vRl9wLUYsNiVRI2VfRidGL0YyLUZTNiYtRiM2J0ZnbkZbb0ZXRmduRjlGOUZmb0Zpby1GNjYtUSI7RidGOUY7Rl5vRkBGQkZERkZGSEZfb0ZNLyUrZXhlY3V0YWJsZUdGPUY5LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYtLUkjbWlHRiQ2JVEjZ3RGJy8lJ2l0YWxpY0dRJXRydWVGJy8lLG1hdGh2YXJpYW50R1EnaXRhbGljRictSSNtb0dGJDYtUSomY29sb25lcTtGJy9GM1Enbm9ybWFsRicvJSZmZW5jZUdRJmZhbHNlRicvJSpzZXBhcmF0b3JHRj0vJSlzdHJldGNoeUdGPS8lKnN5bW1ldHJpY0dGPS8lKGxhcmdlb3BHRj0vJS5tb3ZhYmxlbGltaXRzR0Y9LyUnYWNjZW50R0Y9LyUnbHNwYWNlR1EsMC4yNzc3Nzc4ZW1GJy8lJ3JzcGFjZUdGTC1GLDYlUSdEZ2FtbWFGJ0YvRjItSShtZmVuY2VkR0YkNiYtRiM2Jy1GNjYtUSJ8aXJGJ0Y5RjtGPkZARkJGREZGRkgvRktRLDAuMTExMTExMWVtRicvRk5GZW4tSSNtbkdGJDYkUSI0RidGOS1GNjYtUSIsRidGOUY7L0Y/RjFGQEZCRkRGRkZIL0ZLUSYwLjBlbUYnL0ZOUSwwLjMzMzMzMzNlbUYnLUYsNiVRJ21hdHJpeEYnRi9GMkY5RjkvJSVvcGVuR1EiW0YnLyUmY2xvc2VHUSJdRictRjY2LVEiLkYnRjlGO0Y+RkBGQkZERkZGSEZfby9GTkZgby1GNjYtUSJ+RidGOUY7Rj5GQEZCRkRGRkZIRl9vRl9wLUYsNiVRI2VfRidGL0YyLUZTNiYtRiM2JkZnbkZbb0ZnbkY5RjlGZm9GaW8tRjY2LVEiO0YnRjlGO0Zeb0ZARkJGREZGRkhGX29GTS8lK2V4ZWN1dGFibGVHRj1GOQ==Now the commutations of the above needed for the present calculation are done. Up to a constant factor and i these are often denoted 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 in the 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Notation gamma one index Dirac gamma in Dirac representation, gamma three tetrad indices Ricci rotation symbol, gamma two tetrad indices and one spacetime index the spin connection, which is the same as the Ricci rotation with the tetrad replacing the last tetrad index with a spacetime one. The spin connection is often notated LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkobXN1YnN1cEdGJDYoLUkjbWlHRiQ2JlEoJm9tZWdhO0YnLyUnaXRhbGljR1EmZmFsc2VGJy8lK2V4ZWN1dGFibGVHRjQvJSxtYXRodmFyaWFudEdRJ25vcm1hbEYnLUYjNiUtRi82JlElJm11O0YnRjJGNUY3RjVGNy1GIzYlLUYvNiZRI2FiRicvRjNRJXRydWVGJ0Y1L0Y4USdpdGFsaWNGJ0Y1RjcvJTFzdXBlcnNjcmlwdHNoaWZ0R1EiMEYnLyUvc3Vic2NyaXB0c2hpZnRHRkovSSttc2VtYW50aWNzR0YkUSxbbm9uZSxub25lXUYnLUYvNiNRIUYnRjVGNw== , in the present case maple recognizes the Ricci rotation coefficient with one spacetime index so it is unnecessary to introduce a new symbol.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 the spin connection term \316\223LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUkjbWlHRiQ2JlEnJiM5NTY7RicvJSdpdGFsaWNHUSZmYWxzZUYnLyUrZXhlY3V0YWJsZUdGMS8lLG1hdGh2YXJpYW50R1Enbm9ybWFsRidGMkY0 is calculated, up to a constant it is spin connection LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkobXN1YnN1cEdGJDYoLUkjbWlHRiQ2JlEoJm9tZWdhO0YnLyUnaXRhbGljR1EmZmFsc2VGJy8lK2V4ZWN1dGFibGVHRjQvJSxtYXRodmFyaWFudEdRJ25vcm1hbEYnLUYjNiUtRi82JlElJm11O0YnRjJGNUY3RjVGNy1GIzYlLUYvNiZRI2FiRicvRjNRJXRydWVGJ0Y1L0Y4USdpdGFsaWNGJ0Y1RjcvJTFzdXBlcnNjcmlwdHNoaWZ0R1EiMEYnLyUvc3Vic2NyaXB0c2hpZnRHRkovSSttc2VtYW50aWNzR0YkUSxbbm9uZSxub25lXUYnLUYvNiNRIUYnRjVGNw== times LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUklbXN1YkdGJDYlLUkjbWlHRiQ2JlEoJnNpZ21hO0YnLyUnaXRhbGljR1EmZmFsc2VGJy8lK2V4ZWN1dGFibGVHRjQvJSxtYXRodmFyaWFudEdRJ25vcm1hbEYnLUYjNiUtRi82JlEjYWJGJy9GM1EldHJ1ZUYnRjUvRjhRJ2l0YWxpY0YnRjVGNy8lL3N1YnNjcmlwdHNoaWZ0R1EiMEYnLUkjbW9HRiQ2LlEifkYnRjVGNy8lJmZlbmNlR0Y0LyUqc2VwYXJhdG9yR0Y0LyUpc3RyZXRjaHlHRjQvJSpzeW1tZXRyaWNHRjQvJShsYXJnZW9wR0Y0LyUubW92YWJsZWxpbWl0c0dGNC8lJ2FjY2VudEdGNC8lJ2xzcGFjZUdRJjAuMGVtRicvJSdyc3BhY2VHRlpGNUY3 the commutation of Dirac gamma 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 partial derivative LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYoLUkjbWlHRiQ2J1ErJlBhcnRpYWxEO0YnLyUnaXRhbGljR1EmZmFsc2VGJy8lK2V4ZWN1dGFibGVHRjEvJSxtYXRodmFyaWFudEdRJ25vcm1hbEYnL0krbXNlbWFudGljc0dGJFEqUGh5c2ljc2RfRictSSNtb0dGJDYuUSJ+RidGMkY0LyUmZmVuY2VHRjEvJSpzZXBhcmF0b3JHRjEvJSlzdHJldGNoeUdGMS8lKnN5bW1ldHJpY0dGMS8lKGxhcmdlb3BHRjEvJS5tb3ZhYmxlbGltaXRzR0YxLyUnYWNjZW50R0YxLyUnbHNwYWNlR1EmMC4wZW1GJy8lJ3JzcGFjZUdGTi1GLDYmUScmIzk1NjtGJ0YvRjJGNEY6RjJGNA==and the spin connection term \316\223LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUkjbWlHRiQ2JlEnJiM5NTY7RicvJSdpdGFsaWNHUSZmYWxzZUYnLyUrZXhlY3V0YWJsZUdGMS8lLG1hdGh2YXJpYW50R1Enbm9ybWFsRidGMkY0LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUkjbW9HRiQ2LVEifkYnLyUsbWF0aHZhcmlhbnRHUSdub3JtYWxGJy8lJmZlbmNlR1EmZmFsc2VGJy8lKnNlcGFyYXRvckdGNC8lKXN0cmV0Y2h5R0Y0LyUqc3ltbWV0cmljR0Y0LyUobGFyZ2VvcEdGNC8lLm1vdmFibGVsaW1pdHNHRjQvJSdhY2NlbnRHRjQvJSdsc3BhY2VHUSYwLjBlbUYnLyUncnNwYWNlR0ZDLyUrZXhlY3V0YWJsZUdGNEYvact on a four entry column vector. Define the Dirac column wavefunctionDW(t):=Vector[row]([psi1(t),psi2(t),psi3(t),psi4(t)]):DS(t):=Transpose(DW(t)):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The wavefunctions covariant derivative 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 we can work out the Dirac operator, which is just the Dirac gamma left multiplying the covariant derivative.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JSFHmaple simplifies to the last expressionthe answer should be 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 good news sign and LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUkmbWZyYWNHRiQ2KC1JI21uR0YkNiVRIjNGJy8lK2V4ZWN1dGFibGVHUSZmYWxzZUYnLyUsbWF0aHZhcmlhbnRHUSdub3JtYWxGJy1GIzYmLUYvNiVRIjJGJ0YyRjUvJSdpdGFsaWNHUSV0cnVlRidGMi9GNlEnaXRhbGljRicvJS5saW5ldGhpY2tuZXNzR1EiMUYnLyUrZGVub21hbGlnbkdRJ2NlbnRlckYnLyUpbnVtYWxpZ25HRkcvJSliZXZlbGxlZEdGNEYyRjU= correct, bad news factors of a.LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2I1EhRic=