Return to Answer

It seems OP's main question concerns the systematics of gauge-fixing. We interpret/reformulate OP's questions as essentially the following.

The original gauge-invariant action $S_0$ is unsuitable for quantization, so we add a non-gauge invariant gauge-fixing term to the action. Obviously we cannot add any non-gauge invariant term to the action.

 

1. What is the principle that dictates which gauge-fixing terms are allowed and which are not?

2. And how can the theory be independent of gauge-fixing?

Why we need gauge-fixing and Faddeev-Popov ghosts in the first place were explained in my Phys.SE answer here.

Now, to not get bogged down by technical details, it is actually more convenient to use the BRST formalism. Recall that the BRST formalism is a modern generalization of the Gupta-Bleuler formalism. The BRST transformation $\delta$ basically encodes the gauge transformations.

Moreover, recall that the BRST transformation $\delta$ is Grassmann-odd and nilpotent $\delta^2=0$, and that the original action $S_0$ is BRST-invariant $\delta S_0=0$. The total/gauge-fixed action

$$\tag{1} S_{\rm gf}~=~S_0+\delta\psi$$

is the original action $S_0$ plus a BRST-exact term $\delta\psi$ that depends on the so-called gauge-fixing fermion $\psi$, which encodes the gauge-fixing condition. In other words: different gauge-fixing means different $\psi$.

Note that while the gauge-fixed action $S_{\rm gf}$ is no longer gauge invariant, it is still BRST-invariant.

The BRST-exact term $\delta\psi$ in the action (1) contains both the Faddeev-Popov ghost term and the gauge-fixing terms. Not everything goes: There is an intricate balance between the various terms to ensure that we have only modified the original action $S_0$ with a BRST-exact amount $\delta\psi$, which cannot alter the BRST cohomology, and thereby, in turn, cannot alter the notion of physical states.

This basically answers question 1 and 2 at the conceptional level. For more details, see also e.g. my Phys.SE answer here.

Phrased differently, without the use of the BRST formalism: The usual Faddeev Popov trick (cf. e.g. Ref. 1) precisely codifies the balance between the Faddeev-Popov ghost term and the gauge-fixing terms. They go hand in hand. In the simplest situations, the Faddeev-Popov ghosts decouples, and can be integrated out.

References:

1. M.E. Peskin & D.V. Schroeder, An Intro to QFT, Section 9.4.

It seems OP's main question concerns the systematics of gauge-fixing. We interpret/reformulate OP's questions as essentially the following.

The original gauge-invariant action $S_0$ is unsuitable for quantization, so we add a non-gauge invariant gauge-fixing term to the action. Obviously we cannot add any non-gauge invariant term to the action.

1. What is the principle that dictates which gauge-fixing terms are allowed and which are not?

2. And how can the theory be independent of gauge-fixing?

Why we need gauge-fixing and Faddeev-Popov ghosts in the first place were explained in my Phys.SE answer here.

Now, to not get bogged down by technical details, it is actually more convenient to use the BRST formalism. Recall that the BRST formalism is a modern generalization of the Gupta-Bleuler formalism. The BRST transformation $\delta$ basically encodes the gauge transformations.

Moreover, recall that the BRST transformation $\delta$ is Grassmann-odd and nilpotent $\delta^2=0$, and that the original action $S_0$ is BRST-invariant $\delta S_0=0$. The total/gauge-fixed action

$$\tag{1} S_{\rm gf}~=~S_0+\delta\psi$$

is the original action $S_0$ plus a BRST-exact term $\delta\psi$ that depends on the so-called gauge-fixing fermion $\psi$, which encodes the gauge-fixing condition. In other words: different gauge-fixing means different $\psi$.

Note that while the gauge-fixed action $S_{\rm gf}$ is no longer gauge invariant, it is still BRST-invariant.

The BRST-exact term $\delta\psi$ in the action (1) contains both the Faddeev-Popov ghost term and the gauge-fixing terms. Not everything goes: There is an intricate balance between the various terms to ensure that we have only modified the original action $S_0$ with a BRST-exact amount $\delta\psi$, which cannot alter the BRST cohomology, and thereby, in turn, cannot alter the notion of physical states.

This basically answers question 1 and 2 at the conceptional level. For more details, see also e.g. my Phys.SE answer here.

Phrased differently, without the use of the BRST formalism: The usual Faddeev Popov trick (cf. e.g. Ref. 1) precisely codifies the balance between the Faddeev-Popov ghost term and the gauge-fixing terms. They go hand in hand. In the simplest situations, the Faddeev-Popov ghosts decouples, and can be integrated out.

References:

1. M.E. Peskin & D.V. Schroeder, An Intro to QFT, Section 9.4.

It seems OP's main question concerns the systematics of gauge-fixing. We interpret/reformulate OP's questions as essentially the following.

The original gauge-invariant action $S_0$ is unsuitable for quantization, so we add a non-gauge invariant gauge-fixing term to the action. Obviously we cannot add any non-gauge invariant term to the action.

1. What is the principle that dictates which gauge-fixing terms are allowed and which are not?

2. And how can the theory be independent of gauge-fixing?

Why we need gauge-fixing and Faddeev-Popov ghosts in the first place were explained in my Phys.SE answer here.

Now, to not get bogged down by technical details, it is actually more convenient to use the BRST formalism. Recall that the BRST formalism is a modern generalization of the Gupta-Bleuler formalism. The BRST transformation $\delta$ basically encodes the gauge transformations.

Moreover, recall that the BRST transformation $\delta$ is Grassmann-odd and nilpotent $\delta^2=0$, and that the original action $S_0$ is BRST-invariant $\delta S_0=0$. The total/gauge-fixed action

$$\tag{1} S_{\rm gf}~=~S_0+\delta\psi$$

is the original action $S_0$ plus a BRST-exact term $\delta\psi$ that depends on the so-called gauge-fixing fermion $\psi$, which encodes the gauge-fixing condition. In other words: different gauge-fixing means different $\psi$.

Note that while the gauge-fixed action $S_{\rm gf}$ is no longer gauge invariant, it is still BRST-invariant.

The BRST-exact term $\delta\psi$ in the action (1) contains both the Faddeev-Popov ghost term and the gauge-fixing terms. Not everything goes: There is an intricate balance between the various terms to ensure that we have only modified the original action $S_0$ with a BRST-exact amount $\delta\psi$, which cannot alter the BRST cohomology, and thereby, in turn, cannot alter the notion of physical states.

This basically answers question 1 and 2 at the conceptional level. For more details, see also e.g. my Phys.SE answer here.

Phrased differently, without the use of the BRST formalism: The usual Faddeev Popov trick (cf. e.g. Ref. 1) precisely codifies the balance between the Faddeev-Popov ghost term and the gauge-fixing terms. They go hand in hand. In the simplest situations, the Faddeev-Popov ghosts decouples, and can be integrated out.

References:

1. M.E. Peskin & D.V. Schroeder, An Intro to QFT, Section 9.4.

It seems OP's main question concerns the systematics of gauge-fixing. We interpret/reformulate OP's questions as essentially the following.

The original gauge-invariant action $S_0$ is unsuitable for quantization, so we add a non-gauge invariant gauge-fixing term to the action. Obviously we cannot add any non-gauge invariant term to the action.

1. What is the principle that dictates which gauge-fixing terms are allowed and which are not?

2. And how can the theory be independent of gauge-fixing?

Why we need gauge-fixing and Faddeev-Popov ghosts in the first place were explained in my Phys.SE answer here.

Now, to not get bogged down by technical details, it is actually more convenient to use the BRST formalism. Recall that the BRST formalism is a modern generalization of the Gupta-Bleuler formalism. The BRST transformation $\delta$ basically encodes the gauge transformations.

Moreover, recall that the BRST transformation $\delta$ is Grassmann-odd and nilpotent $\delta^2=0$, and that the original action $S_0$ is BRST-invariant $\delta S_0=0$. The total/gauge-fixed action

$$\tag{1} S_{\rm gf}~=~S_0+\delta\psi$$

is the original action $S_0$ plus a BRST-exact term $\delta\psi$ that depends on the so-called gauge-fixing fermion $\psi$, which encodes the gauge-fixing condition. In other words: different gauge-fixing means different $\psi$.

Note that while the gauge-fixed action $S_{\rm gf}$ is no longer gauge invariant, it is still BRST-invariant.

The BRST-exact term $\delta\psi$ in the action (1) contains both the Faddeev-Popov ghost term and the gauge-fixing terms. Not everything goes: There is an intricate balance between the various terms to ensure that we have only modified the original action $S_0$ with a BRST-exact amount $\delta\psi$, which cannot alter the BRST cohomology, and thereby, in turn, cannot alter the notion of physical states.

This basically answers question 1 and 2 at the conceptional level. For more details, see also e.g. my Phys.SE answer here.

Phrased differently, without the use of the BRST formalism: The usual Faddeev Popov trick (cf. e.g. Ref. 1) precisely codifies the balance between the Faddeev-Popov ghost term and the gauge-fixing terms. They go hand in hand. In the simplest situations, the Faddeev-Popov ghosts decouples, and can be integrated out.

References:

1. M.E. Peskin & D.V. Schroeder, An Intro to QFT, Section 9.4.

It seems OP's main question concerns the systematics of gauge-fixing. We reformulate OP's main question as follows:

The original gauge-invariant action $S_0$ is unsuitable for quantization, so we add a non-gauge invariant gauge-fixing term to the action. Obviously we cannot add any non-gauge invariant term to the action.

1. What is the principle that dictates which gauge-fixing terms are allowed and which are not?

2. And how can the theory be independent of gauge-fixing?

Why we need gauge-fixing and Faddeev-Popov ghosts in the first place were explained in my Phys.SE answer here.

Now, to not get bogged down by technical details, it is actually more convenient to use the BRST formalism. Recall that BRST formalism is a modern generalization of the Gupta-Bleuler formalism. The BRST transformation $\delta$ basically encodes the gauge transformations.

Moreover, recall that the BRST transformation $\delta$ is Grassmann-odd and nilpotent $\delta^2=0$, and the original action $S_0$ is BRST-invariant $\delta S_0=0$. The total, gauge-fixed action

$$S_{\rm gf}~=~S_0+\delta\psi$$

is the original action $S_0$ plus a BRST-exact term $\delta\psi$ that depends on the so-called gauge-fixing fermion $\psi$, which encodes the gauge-fixing condition.

The BRST-exact term $\delta\psi$ contains both the Faddeev-Popov ghost term and the gauge-fixing terms. Not everythings goes. There is an intricate balance between the various terms to ensure that we have only modified the original action $S_0$ with a BRST-exact amount $\delta\psi$, which cannot alter the BRST cohomology, and thereby, in turn, the notion of physical states. This basically answers question 1 and 2 at the conceptional level. For more details, see also my Phys.SE answer here.

Phrased differently, without the use of the BRST formalism: The usual Faddeev Popov trick (cf. e.g. Ref. 1) precisely codifies the balance between the Faddeev-Popov ghost term and the gauge-fixing terms. They go hand in hand. In the simplest situations, the Faddeev-Popov ghosts decouples, and can be integrated out.

References:

1. M.E. Peskin & D.V. Schroeder, An Intro to QFT, Section 9.4.

It seems OP's main question concerns the systematics of gauge-fixing. We interpret/reformulate OP's questions as essentially the following.

The original gauge-invariant action $S_0$ is unsuitable for quantization, so we add a non-gauge invariant gauge-fixing term to the action. Obviously we cannot add any non-gauge invariant term to the action.

1. What is the principle that dictates which gauge-fixing terms are allowed and which are not?

2. And how can the theory be independent of gauge-fixing?

Why we need gauge-fixing and Faddeev-Popov ghosts in the first place were explained in my Phys.SE answer here.

Now, to not get bogged down by technical details, it is actually more convenient to use the BRST formalism. Recall that the BRST formalism is a modern generalization of the Gupta-Bleuler formalism. The BRST transformation $\delta$ basically encodes the gauge transformations.

Moreover, recall that the BRST transformation $\delta$ is Grassmann-odd and nilpotent $\delta^2=0$, and that the original action $S_0$ is BRST-invariant $\delta S_0=0$. The total/gauge-fixed action

$$\tag{1} S_{\rm gf}~=~S_0+\delta\psi$$

is the original action $S_0$ plus a BRST-exact term $\delta\psi$ that depends on the so-called gauge-fixing fermion $\psi$, which encodes the gauge-fixing condition. In other words: different gauge-fixing means different $\psi$.

Note that while the gauge-fixed action $S_{\rm gf}$ is no longer gauge invariant, it is still BRST-invariant.

The BRST-exact term $\delta\psi$ in the action (1) contains both the Faddeev-Popov ghost term and the gauge-fixing terms. Not everything goes: There is an intricate balance between the various terms to ensure that we have only modified the original action $S_0$ with a BRST-exact amount $\delta\psi$, which cannot alter the BRST cohomology, and thereby, in turn, cannot alter the notion of physical states. 

This basically answers question 1 and 2 at the conceptional level. For more details, see also e.g. my Phys.SE answer here.

Phrased differently, without the use of the BRST formalism: The usual Faddeev Popov trick (cf. e.g. Ref. 1) precisely codifies the balance between the Faddeev-Popov ghost term and the gauge-fixing terms. They go hand in hand. In the simplest situations, the Faddeev-Popov ghosts decouples, and can be integrated out.

References:

1. M.E. Peskin & D.V. Schroeder, An Intro to QFT, Section 9.4.