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$\gamma_a$ lives in the spin/Clifford bundle denoted by Roman indices $a$, while spacetime $dx^\mu$ is characterized by Greek indices $\mu$.

Romans and Greeks shake hands with the help of tetrad (or vierbein) $$ e^a_\mu \gamma_a dx^\mu $$ The flat Minkowsky space (with metric $g_{\mu\nu} = \eta_{ab}e^a_\mu e^a_\nu = \eta_{\mu\nu}$) is characterized by $$ e^a_\mu = \delta^a_\mu $$ Therefore $$ e^a_\mu \gamma_a dx^\mu = \delta^a_\mu\gamma_a dx^\mu = \gamma_\mu dx^\mu $$ The delta function $\delta^a_\mu$ is kinda of "soldering" the Roman and Greek indices together, hence the perfect linkage between the fundamental (4-vector) representation and the bispinor (sandwiching $\gamma_\mu$ between two spinors $\bar{\psi}\gamma_\mu\psi$) representations.

In curved spacetime, $e^a_\mu = \delta^a_\mu$ is not true anymore. Therefore you have to be fluent in both Roman and Greek languages and use $e^a_\mu$ as a dictionary.

$\gamma_a$ lives in the spin/Clifford bundle denoted by Roman indices $a$, while spacetime $dx^\mu$ is characterized by Greek indices $\mu$.

Romans and Greeks shake hands with the help of tetrad (or vierbein) $$ e^a_\mu \gamma_a dx^\mu $$ The flat Minkowsky space (with metric $g_{\mu\nu} = \eta_{ab}e^a_\mu e^a_\nu = \eta_{\mu\nu}$) is characterized by $$ e^a_\mu = \delta^a_\mu $$ Therefore $$ e^a_\mu \gamma_a dx^\mu = \delta^a_\mu\gamma_a dx^\mu = \gamma_\mu dx^\mu $$ The delta function $\delta^a_\mu$ is kinda of "soldering" the Roman and Greek indices together, hence the perfect linkage between the fundamental (4-vector) representation and the bispinor (sandwiching $\gamma_\mu$ between two spinors $\bar{\psi}\gamma_\mu\psi$) representations.

In curved spacetime, the simple correspondence $e^a_\mu = \delta^a_\mu$ is not true anymore. Therefore you have to be fluent in both Roman and Greek languages and use $e^a_\mu$ as a dictionary.

$\gamma_a$ lives in the spin/Clifford bundle denoted by Roman indices $a$, while spacetime $dx^\mu$ is characterized by Greek indices $\mu$.

Romans and Greeks shake hands with the help of tetrad (or vierbein) $$ e^a_\mu \gamma_a dx^\mu $$ The flat Minkowsky space (with metric $g_{\mu\nu} = \eta_{ab}e^a_\mu e^a_\nu = \eta_{\mu\nu}$) is characterized by $$ e^a_\mu = \delta^a_\mu $$ Therefore $$ e^a_\mu \gamma_a dx^\mu = \delta^a_\mu\gamma_a dx^\mu = \gamma_\mu dx^\mu $$ The delta function $\delta^a_\mu$ is kinda of "soldering" the Roman and Greek indices together, hence the perfect linkage between the fundamental (4-vector) representation and the bispinor (sandwiching $\gamma_\mu$ between two spinors $\bar{\psi}\gamma_\mu\psi$) representations.

In curved spacetime, $e^a_\mu = \delta^a_\mu$ is not true anymore. Therefore you have to be friends with both Romans and Greeks and use $e^a_\mu$ to grease the cross-border transactions.

$\gamma_a$ lives in the spin/Clifford bundle denoted by Roman indices $a$, while spacetime $dx^\mu$ is characterized by Greek indices $\mu$.

Romans and Greeks shake hands with the help of tetrad (or vierbein) $$ e^a_\mu \gamma_a dx^\mu $$ The flat Minkowsky space (with metric $g_{\mu\nu} = \eta_{ab}e^a_\mu e^a_\nu = \eta_{\mu\nu}$) is characterized by $$ e^a_\mu = \delta^a_\mu $$ Therefore $$ e^a_\mu \gamma_a dx^\mu = \delta^a_\mu\gamma_a dx^\mu = \gamma_\mu dx^\mu $$ The delta function $\delta^a_\mu$ is kinda of "soldering" the Roman and Greek indices together, hence the perfect linkage between the fundamental (4-vector) representation and the bispinor (sandwiching $\gamma_\mu$ between two spinors $\bar{\psi}\gamma_\mu\psi$) representations.

In curved spacetime, $e^a_\mu = \delta^a_\mu$ is not true anymore. Therefore you have to be fluent in both Roman and Greek languages and use $e^a_\mu$ as a dictionary.

$\gamma_a$ lives in the spin/Clifford bundle denoted by roman indices $a$, while spacetime $dx^\mu$ is characterized by Greek indices $\mu$.

Romans and Greeks shake hands with the help of tetrad (or vierbein) $$ e^a_\mu \gamma_a dx^\mu $$ The flat Minkowsky space (with metric $g_{\mu\nu} = \eta_{ab}e^a_\mu e^a_\nu = \eta_{\mu\nu}$) is characterized by $$ e^a_\mu = \delta^a_\mu $$ Therefore $$ e^a_\mu \gamma_a dx^\mu = \delta^a_\mu\gamma_a dx^\mu = \gamma_\mu dx^\mu $$ The delta function $\delta^a_\mu$ is kinda of "soldering" the Roman and Greek indices together, hence the linkage between the fundamental (4-vector) representation and the bispinor (sandwiching $\gamma_\mu$ between two spinors $\bar{\psi}\gamma_\mu\psi$) representations.

In curved spacetime, $e^a_\mu = \delta^a_\mu$ is not true anymore. Therefore you have to be friends with both Romans and Greeks and use $e^a_\mu$ to grease the cross-border transactions.

$\gamma_a$ lives in the spin/Clifford bundle denoted by Roman indices $a$, while spacetime $dx^\mu$ is characterized by Greek indices $\mu$.

Romans and Greeks shake hands with the help of tetrad (or vierbein) $$ e^a_\mu \gamma_a dx^\mu $$ The flat Minkowsky space (with metric $g_{\mu\nu} = \eta_{ab}e^a_\mu e^a_\nu = \eta_{\mu\nu}$) is characterized by $$ e^a_\mu = \delta^a_\mu $$ Therefore $$ e^a_\mu \gamma_a dx^\mu = \delta^a_\mu\gamma_a dx^\mu = \gamma_\mu dx^\mu $$ The delta function $\delta^a_\mu$ is kinda of "soldering" the Roman and Greek indices together, hence the perfect linkage between the fundamental (4-vector) representation and the bispinor (sandwiching $\gamma_\mu$ between two spinors $\bar{\psi}\gamma_\mu\psi$) representations.

In curved spacetime, $e^a_\mu = \delta^a_\mu$ is not true anymore. Therefore you have to be friends with both Romans and Greeks and use $e^a_\mu$ to grease the cross-border transactions.