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question: WHY BRST formulation works? In more details:

1. What are the conditions we need to impose on QFT to find the BRST (global) symmetry?
2. Why can we demand the BRST parameter $\epsilon$ directly that relates the gauge symmetry parameter $\alpha^a(x)$ to the ghost field?
3. And how to determine how many BRST parameters $$\epsilon_1,\epsilon_2,\epsilon_3,...$$ can we introduce? Does each continuous BRST parameter $\epsilon_j$ introduce a $U(1)$ or a supergroup (?) global symmetry? (But note that $\epsilon_j$ is an anticommuting Grassman number.) and correspond to a ghost field one by one?

(You can follow discussions on Peskin and Schroeder (PS) Chap 16.4 if you wish, also with this understanding.)

Given a gauge theory such as nonabelian Yang-Mills gauge theory, we know there is a gauge symmetry transformation on the 1-form gauge field as (in the standard QFT notation of spacetime index $\nu$ and the gauge Lie algebra (adjoint) index $a, b, c$) $$ A^a_\nu \to A^a_\nu +\delta A^a_\nu = A^a_\nu +\frac{1}{g}D_\nu^{ac} \alpha^c =A^a_\nu +\frac{1}{g} (\partial_\nu \delta^{ac} + g f^{abc} A^b_\nu ) \alpha^c $$ with a 0-form gauge parameter $\alpha^a$.

However the BRST formulation declares that we can introduce the global symmetry parameter $\epsilon$ and a $C$ ghost field: $$ A^a_\nu \to A^a_\nu +\delta A^a_\nu = A^a_\nu +\epsilon D_\nu^{ac} C^c $$ such that we need a relation between the gauge symmetry and BRST global symmetry: $$ \alpha^a = g \epsilon C^a. $$ (Pardon me here that I capitalize the ghost field $C$ to distinguish it from the gauge index $c$.)

• But how do we know such a "single" BRST parameter can be introduced? (What conditions do we need to impose and declare such global symmetry $\epsilon$?)

• Follow the question 3 above, If we have more ghost fields (such as Chap 2.5 Polchinski) with $b$ and $c$ two ghost fields for $bc$ CFT, is it possible to introduce more BRST parameters $\epsilon_1,\epsilon_2,..$? Is the number of the BRST symmetry parameters the same as the number of ghost fields?

question: WHY BRST formulation works? In more details:

1. What are the conditions we need to impose on QFT to find the BRST (global) symmetry?
2. Why can we demand the BRST parameter $\epsilon$ directly that relates the gauge symmetry parameter $\alpha^a(x)$ to the ghost field?
3. And how to determine how many BRST parameters $$\epsilon_1,\epsilon_2,\epsilon_3,...$$ can we introduce? Does each continuous BRST parameter $\epsilon_j$ introduce a $U(1)$ or a supergroup (?) global symmetry? (But note that $\epsilon_j$ is an anticommuting Grassman number.) and correspond to a ghost field one by one?

(You can follow discussions on Peskin and Schroeder (PS) Chap 16.4 if you wish, also with this understanding.)

Given a gauge theory such as nonabelian Yang-Mills gauge theory, we know there is a gauge symmetry transformation on the 1-form gauge field as (in the standard QFT notation of spacetime index $\nu$ and the gauge Lie algebra (adjoint) index $a, b, c$) $$ A^a_\nu \to A^a_\nu +\delta A^a_\nu = A^a_\nu +\frac{1}{g}D_\nu^{ac} \alpha^c =A^a_\nu +\frac{1}{g} (\partial_\nu \delta^{ac} + g f^{abc} A^b_\nu ) \alpha^c $$ with a 0-form gauge parameter $\alpha^a$.

However the BRST formulation declares that we can introduce the global symmetry parameter $\epsilon$ and a $C$ ghost field: $$ A^a_\nu \to A^a_\nu +\delta A^a_\nu = A^a_\nu +\epsilon D_\nu^{ac} C^c $$ such that we need a relation between the gauge symmetry and BRST global symmetry: $$ \boxed{\alpha^a = g \epsilon C^a}. $$ (Pardon me here that I capitalize the ghost field $C$ to distinguish it from the gauge index $c$.)

• But how do we know such a "single" BRST parameter can be introduced? (What conditions do we need to impose and declare such global symmetry $\epsilon$?)

• Follow the question 3 above, If we have more ghost fields (such as Chap 2.5 Polchinski) with $b$ and $c$ two ghost fields for $bc$ CFT, is it possible to introduce more BRST parameters $\epsilon_1,\epsilon_2,..$? Is the number of the BRST symmetry parameters the same as the number of ghost fields?

question: WHY BRST formulation works? In more details:

1. What are the conditions we need to impose on QFT to find the BRST (global) symmetry?
2. Why can we demand the BRST parameter $\epsilon$ directly that relates the gauge symmetry parameter $\alpha^a(x)$ to the ghost field?
3. And how to determine how many BRST parameters $\epsilon_1,\epsilon_2,\epsilon_3,...$ can we introduce? Does each continuous BRST parameter $\epsilon_j$ introduce a $U(1)$ or a supergroup (?) global symmetry? (But note that $\epsilon_j$ is an anticommuting Grassman number.)

(You can follow discussions on Peskin and Schroeder (PS) Chap 16.4 if you wish, also with this understanding.)

Given a gauge theory such as nonabelian Yang-Mills gauge theory, we know there is a gauge symmetry transformation on the 1-form gauge field as (in the standard QFT notation of spacetime index $\nu$ and the gauge Lie algebra (adjoint) index $a, b, c$) $$ A^a_\nu \to A^a_\nu +\delta A^a_\nu = A^a_\nu +\frac{1}{g}D_\nu^{ac} \alpha^c =A^a_\nu +\frac{1}{g} (\partial_\nu \delta^{ac} + g f^{abc} A^b_\nu ) \alpha^c $$ with a 0-form gauge parameter $\alpha^a$.

However the BRST formulation declares that we can introduce the global symmetry parameter $\epsilon$ and a $C$ ghost field: $$ A^a_\nu \to A^a_\nu +\delta A^a_\nu = A^a_\nu +\epsilon D_\nu^{ac} C^c $$ such that we need a relation between the gauge symmetry and BRST global symmetry: $$ \alpha^a = g \epsilon C^a. $$ (Pardon me that I capitalize the ghost field $C$ to distinguish it from the gauge index $c$.)

• But how do we know such a "single" BRST parameter can be introduced? (What conditions do we need to impose and declare such global symmetry $\epsilon$?)

• Follow the question 3 above, If we have more ghost fields (such as Chap 2.5 Polchinski) with $b$ and $c$ two ghost fields for $bc$ CFT, is it possible to introduce more BRST parameters $\epsilon_1,\epsilon_2,..$? Is the number of the BRST symmetry parameters the same as the number of ghost fields?

question: WHY BRST formulation works? In more details:

1. What are the conditions we need to impose on QFT to find the BRST (global) symmetry?
2. Why can we demand the BRST parameter $\epsilon$ directly that relates the gauge symmetry parameter $\alpha^a(x)$ to the ghost field?
3. And how to determine how many BRST parameters $$\epsilon_1,\epsilon_2,\epsilon_3,...$$ can we introduce? Does each continuous BRST parameter $\epsilon_j$ introduce a $U(1)$ or a supergroup (?) global symmetry?   (But note that $\epsilon_j$ is an anticommuting Grassman number.) and correspond to a ghost field one by one?

(You can follow discussions on Peskin and Schroeder (PS) Chap 16.4 if you wish, also with this understanding.)

Given a gauge theory such as nonabelian Yang-Mills gauge theory, we know there is a gauge symmetry transformation on the 1-form gauge field as (in the standard QFT notation of spacetime index $\nu$ and the gauge Lie algebra (adjoint) index $a, b, c$) $$ A^a_\nu \to A^a_\nu +\delta A^a_\nu = A^a_\nu +\frac{1}{g}D_\nu^{ac} \alpha^c =A^a_\nu +\frac{1}{g} (\partial_\nu \delta^{ac} + g f^{abc} A^b_\nu ) \alpha^c $$ with a 0-form gauge parameter $\alpha^a$.

However the BRST formulation declares that we can introduce the global symmetry parameter $\epsilon$ and a $C$ ghost field: $$ A^a_\nu \to A^a_\nu +\delta A^a_\nu = A^a_\nu +\epsilon D_\nu^{ac} C^c $$ such that we need a relation between the gauge symmetry and BRST global symmetry: $$ \alpha^a = g \epsilon C^a. $$ (Pardon me here that I capitalize the ghost field $C$ to distinguish it from the gauge index $c$.)

• But how do we know such a "single" BRST parameter can be introduced? (What conditions do we need to impose and declare such global symmetry $\epsilon$?)

• Follow the question 3 above, If we have more ghost fields (such as Chap 2.5 Polchinski) with $b$ and $c$ two ghost fields for $bc$ CFT, is it possible to introduce more BRST parameters $\epsilon_1,\epsilon_2,..$? Is the number of the BRST symmetry parameters the same as the number of ghost fields?

question: WHY BRST formulation works? In more details:

1. What are the conditions we need to impose on QFT to find the BRST (global) symmetry?
2. Why can we demand the BRST parameter $\epsilon$ directly that relates the gauge symmetry parameter $\alpha^a(x)$ to the ghost field?
3. And how to determine how many BRST parameters $\epsilon_1,\epsilon_2,\epsilon_3,...$ can we introduce? Does each continuous BRST parameter $\epsilon_j$ introduce a $U(1)$ or a supergroup (?) global symmetry? (But note that $\epsilon_j$ is an anticommuting Grassman number.)

(You can follow discussions on Peskin and Schroeder (PS) Chap 16.4 if you wish, also with this understanding.)

Given a gauge theory such as nonabelian Yang-Mills gauge theory, we know there is a gauge symmetry transformation on the 1-form gauge field as (in the standard notation of spacetime $\nu$ and the gauge Lie algebra (adjoint) index $a, b, c$) $$ A^a_\nu \to A^a_\nu +\delta A^a_\nu = A^a_\nu +\frac{1}{g}D_\nu^{ac} \alpha^c =A^a_\nu +\frac{1}{g} (\partial_\nu \delta^{ac} + g f^{abc} A^b_\nu ) \alpha^c $$ with a 0-form gauge parameter $alpha^a$.

However the BRST formulation declares that we can introduce the global symmetry parameter $\epsilon$ and a $c$ ghost field: $$ A^a_\nu \to A^a_\nu +\delta A^a_\nu = A^a_\nu +\epsilon D_\nu^{ac} \alpha^c $$ such that we need a relation between the gauge symmetry and BRST global symmetry: $$ \alpha^a = g \epsilon c^a. $$

• But how do we know such a "single" BRST parameter can be introduced? (What conditions do we need to impose and declare such global symmetry $\epsilon$.)

• Follow the question 3 above, If we have more ghost fields (such as Chap 2.5 Polchinski) with $b$ and $c$ two ghost fields for $bc$ CFT, is it possible to introduce more BRST parameters $\epsilon_1,\epsilon_2,..$? Is the number of the BRST symmetry parameters the same as the number of ghost fields?

question: WHY BRST formulation works? In more details:

1. What are the conditions we need to impose on QFT to find the BRST (global) symmetry?
2. Why can we demand the BRST parameter $\epsilon$ directly that relates the gauge symmetry parameter $\alpha^a(x)$ to the ghost field?
3. And how to determine how many BRST parameters $\epsilon_1,\epsilon_2,\epsilon_3,...$ can we introduce? Does each continuous BRST parameter $\epsilon_j$ introduce a $U(1)$ or a supergroup (?) global symmetry? (But note that $\epsilon_j$ is an anticommuting Grassman number.)

(You can follow discussions on Peskin and Schroeder (PS) Chap 16.4 if you wish, also with this understanding.)

Given a gauge theory such as nonabelian Yang-Mills gauge theory, we know there is a gauge symmetry transformation on the 1-form gauge field as (in the standard QFT notation of spacetime index $\nu$ and the gauge Lie algebra (adjoint) index $a, b, c$) $$ A^a_\nu \to A^a_\nu +\delta A^a_\nu = A^a_\nu +\frac{1}{g}D_\nu^{ac} \alpha^c =A^a_\nu +\frac{1}{g} (\partial_\nu \delta^{ac} + g f^{abc} A^b_\nu ) \alpha^c $$ with a 0-form gauge parameter $\alpha^a$.

However the BRST formulation declares that we can introduce the global symmetry parameter $\epsilon$ and a $C$ ghost field: $$ A^a_\nu \to A^a_\nu +\delta A^a_\nu = A^a_\nu +\epsilon D_\nu^{ac} C^c $$ such that we need a relation between the gauge symmetry and BRST global symmetry: $$ \alpha^a = g \epsilon C^a. $$ (Pardon me that I capitalize the ghost field $C$ to distinguish it from the gauge index $c$.)

• But how do we know such a "single" BRST parameter can be introduced? (What conditions do we need to impose and declare such global symmetry $\epsilon$?)

• Follow the question 3 above, If we have more ghost fields (such as Chap 2.5 Polchinski) with $b$ and $c$ two ghost fields for $bc$ CFT, is it possible to introduce more BRST parameters $\epsilon_1,\epsilon_2,..$? Is the number of the BRST symmetry parameters the same as the number of ghost fields?