Hamiltonian Gauge Theory

Introduction

Up to this point in the course, gauge invariance has mostly appeared in Lagrangian language. The Maxwell–Dirac theory was introduced as a local $U(1)$-invariant field theory, and that local symmetry determined the form of the electromagnetic interaction. But to build canonical quantum field theory, we need a Hamiltonian.

That is where the trouble begins.

For ordinary unconstrained fields, the Hamiltonian formalism is straightforward: identify the canonical coordinates, compute their conjugate momenta, perform the Legendre transform, and obtain a phase-space dynamics. But for a gauge field, things are different. The gauge potential $A_\mu$ has four components, so naively one might think it gives four coordinates and four momenta, making eight phase-space variables. The lecture’s main point is that this is wrong. Local gauge symmetry removes part of this apparent phase space.

The reason is deep and general. A local symmetry does two things at once:

• it imposes constraints, which remove independent dynamics,
• and it introduces freedoms, which are not determined by the equations of motion or by initial data.

Those two effects are canonically conjugate aspects of the same symmetry. The lecture spends real time on this, because if you do not understand that, gauge Hamiltonians look like arbitrary tricks rather than structured reductions of the original theory.

Once this general point is clear, the lecture applies it to electromagnetism. It shows that:

• one canonical momentum vanishes identically,
• the $A_0$ equation of motion becomes a universal constraint,
• the corresponding gauge freedom remains arbitrary,
• and therefore one cannot write a useful Hamiltonian until one fixes a gauge.

Among the infinitely many possible gauge choices, the lecture chooses Coulomb gauge because it is simple and physically transparent for basic calculations. In that gauge, the vector potential is expanded in transverse polarization vectors, the scalar potential becomes an instantaneous Coulomb integral over the charge density, and the Hamiltonian reduces to one involving only the two transverse dynamical electromagnetic degrees of freedom plus the instantaneous Coulomb interaction.

The lecture then makes one more important conceptual point. No individual Hamiltonian is gauge invariant by itself. Different gauge choices produce different Hamiltonians, though they are canonically equivalent. Likewise, no single Hamiltonian is manifestly Lorentz invariant. This is why the Hamiltonian formalism is both necessary and limited:

• necessary to define the canonical quantum theory,
• limited because it hides the symmetries that make gauge QFT work so well.

That tension is exactly what motivates the later move to path integrals. :contentReference[oaicite:1]{index=1}

Learning Objectives

• Recall the gauge-invariant Maxwell–Dirac Lagrangian density.
• Understand qualitatively how a local symmetry affects Hamiltonian phase space.
• Explain why a local symmetry produces both universal constraints and arbitrary gauge freedoms.
• Count the reduction in dynamical variables caused by a $U(1)$ gauge symmetry.
• Derive the canonical momenta of the electromagnetic field and explain why $\Pi_0 = 0$.
• Understand the $A_0$ Euler–Lagrange equation as a universal constraint.
• Explain why $A_0$ is not an independent dynamical field.
• Understand why gauge fixing is necessary to obtain a practical Hamiltonian formulation.
• Compare several gauge choices and understand why Coulomb gauge is chosen here.
• Expand the vector potential in transverse polarization modes in Fourier space.
• Derive the Coulomb-gauge Maxwell Hamiltonian and identify its physical terms.
• Explain why only the two transverse photon degrees of freedom remain dynamical.
• Understand why different gauge-fixed Hamiltonians are canonically equivalent.
• Explain why Lorentz and gauge invariance are not manifest in the canonical formulation and why path integrals later help restore them.

Prerequisite Knowledge

• Maxwell–Dirac Lagrangian and local $U(1)$ gauge symmetry
• Canonical Hamiltonian field theory
• Euler–Lagrange equations for fields
• Noether theorem
• Basic electrodynamics
• Fourier transforms
• Polarization vectors for electromagnetic waves
• Canonical transformations

1. Review: the Maxwell–Dirac Lagrangian

The lecture begins by recalling the Maxwell–Dirac Lagrangian density. It writes the total Lagrangian as

$$\mathcal L = \mathcal L_A + \mathcal L_D,$$

where the Dirac piece is

$$\mathcal L_D = i\hbar c\,\bar\psi\gamma^\mu \partial_\mu \psi - Mc^2 \bar\psi\psi,$$

and the gauge-field part is the usual electromagnetic term together with the coupling to the Dirac current. The field strength is

$$F_{\mu\nu} = \partial_\mu A_\nu - \partial_\nu A_\mu,$$

and the gauge field $A_\mu$ is identified as the $U(1)$ gauge field, i.e. the electromagnetic 4-potential up to units. The lecture also reminds us that the full Lagrangian is invariant under

$$\psi \to e^{iqf(x,t)/\hbar}\psi, \qquad A_\mu \to A_\mu + \partial_\mu f(x,t),$$

for any real function $f(x,t)$. :contentReference[oaicite:2]{index=2}

This is the starting point. The gauge symmetry is manifest at the Lagrangian level.

2. The goal: a Hamiltonian for electrodynamics

The lecture then states the goal clearly:

$$H = H_A + H_D.$$

It says we already have the Dirac-field Hamiltonian

$$H_D = \int d^3x\, \left( -i\hbar c\, \bar\psi \vec\gamma\cdot \vec D\psi + Mc^2 \bar\psi\psi \right)$$

in the coupled theory, but

$$H_A$$

turns out to be a longer story. :contentReference[oaicite:3]{index=3}

That “longer story” is exactly the content of the lecture: the gauge field cannot simply be Legendre-transformed in the naive way because not all components are independent dynamical variables.

3. Qualitative example: how symmetry changes phase space

Before touching electromagnetism, the lecture gives a simpler mechanical example: a particle in a rotationally symmetric potential,

$$L = \frac{m}{2}(\dot x^2 + \dot y^2) - V(x^2+y^2).$$

Rotational symmetry gives the conserved Noether charge

$$Q = m y\dot x - m x\dot y,$$

the angular momentum. Switching to polar coordinates makes the structure transparent:

• the angle $\theta$ is the symmetry coordinate,
• its canonically conjugate momentum is the conserved angular momentum,
• if $(r(t),\theta(t))$ is a solution, then $(r(t),\theta(t)+\theta_0)$ is also a solution for any constant $\theta_0$,
• but changing $\dot\theta$ is not itself a symmetry. :contentReference[oaicite:4]{index=4}

The lecture uses this to explain a general principle: for every one phase-space coordinate of symmetry, one canonically conjugate pair is affected. One part becomes constrained, and the other part becomes unconstrained.

This is the right conceptual template for gauge theory.

4. Global versus local symmetry in Hamiltonian language

The lecture then explains the difference between global and local symmetry.

For a single global symmetry, there is only one constraint for all time, so the constrained quantity may still depend on the initial conditions. In that sense it remains a dynamical variable. Similarly, the symmetry freedom itself is only one freedom for all time, so it is also ultimately fixed by initial conditions.

For a local symmetry, the situation is much stronger:

• the constraints apply independently at every time,
• therefore the constrained quantities cannot even depend on initial conditions,
• and the gauge freedom is so large that even after fixing all boundary conditions, one may still gauge-transform one solution into another solution with the same boundary conditions.

So in a local symmetry:

• the constrained coordinate becomes universally fixed,
• the gauge coordinate becomes completely arbitrary,
• and neither behaves like an ordinary dynamical variable anymore. :contentReference[oaicite:5]{index=5}

This is one of the most important conceptual parts of the lecture.

5. Counting electromagnetic degrees of freedom

The lecture then applies that reasoning to the electromagnetic field.

Naively, $A_\mu$ gives four field components plus four canonical momenta:

$$A_\mu,\quad \Pi^\mu_A,$$

so one might think there are eight dynamical field variables.

But a one-parameter local symmetry in a field theory corresponds to two phase-space coordinates of symmetry per spatial point, because the gauge parameter and its time derivative can both vary independently. Therefore a $U(1)$ gauge symmetry removes four dynamical field variables.

So the lecture concludes: the $A_\mu$ Hamiltonian should have only four dynamical field variables, not eight. :contentReference[oaicite:6]{index=6}

This matches physical intuition:

• two canonical coordinate-momentum pairs,
• corresponding to the two transverse photon polarizations.

6. First universal constraint: $\Pi_0 = 0$

The lecture then computes the canonical momentum fields of the electromagnetic part of the Lagrangian.

For the spatial components,

$$\Pi_i = \frac{\partial \mathcal L_A}{\partial \dot A_i},$$

and this is proportional to the electric field, as expected.

But for the time component,

$$\Pi_0 = \frac{\partial \mathcal L_A}{\partial \dot A_0} = 0.$$

The lecture explicitly labels this as the first universal constraint, unaffected by initial conditions. :contentReference[oaicite:7]{index=7}

This is the canonical sign that $A_0$ is not an ordinary dynamical field coordinate. There is no independent conjugate momentum for it because the Lagrangian contains no $\dot A_0$.

7. Second universal constraint: the $A_0$ field equation

The next universal constraint comes from the Euler–Lagrange equation for $A_0$. The lecture shows that this equation determines the Laplacian of $A_0$ in terms of the charge density and the time derivative of the divergence of $\vec A$. Solving it gives

$$A_0(\vec x,t) = -\frac{1}{4\pi} \int d^3x'\, \frac{j^0(\vec x',t) - \frac{1}{c}\nabla\cdot \dot{\vec A}(\vec x',t)} {|\vec x' - \vec x|} + F(\vec x,t),$$

for any function $F$ satisfying

$$\nabla^2 F = 0.$$

The lecture emphasizes what this means:

• the Laplacian of $A_0$ is universally constrained,
• once the other dynamical variables are known, there is no independent evolution left for $A_0$,
• but $A_0$ is still not uniquely fixed, because the harmonic function $F$ remains arbitrary.

So $A_0$ is partly constrained and partly gauge freedom. The lecture then says we can eliminate $A_0$ from the Lagrangian by replacing it with this solution. In that sense, the two constrained coordinates $A_0$ and $\Pi_0$ effectively disappear from $\mathcal L$. :contentReference[oaicite:8]{index=8}

This is the second key reduction.

8. The two arbitrary components: residual gauge freedom

Even after eliminating $A_0$ and $\Pi_0$, there are still two parameters’ worth of total freedom left in the gauge sector.

The lecture writes the gauge transformation of the spatial vector potential and its time derivative:

$$A_i \to A_i + \partial_i f, \qquad \dot A_i \to \dot A_i + \partial_i \dot f.$$

So there are still two dynamical variables’ worth of symmetry. Their evolution is arbitrary. That means one cannot get a Hamiltonian dynamics for them directly. Instead, one must pick an arbitrary choice for the gauge degrees of freedom, and then put the remaining variables into Hamiltonian form. :contentReference[oaicite:9]{index=9}

The lecture phrases this sharply: determining two more degrees of freedom by arbitrary choice means imposing two more universal constraints, independent of initial conditions — except now it is up to us what those constraints are.

That is gauge fixing.

9. Possible gauges

The lecture then lists several possible gauge choices.

Examples include:

• axial gauge, such as $A_3=A_0=0$,
• Lorentz gauge, $\partial_\mu A^\mu = 0$,
• Coulomb gauge, $\nabla\cdot \vec A = 0$.

It says there are infinitely many possible choices, but chooses Coulomb gauge because it is especially good for low-energy problems and for the next part of the course. :contentReference[oaicite:10]{index=10}

This is an important practical choice, not a claim that Coulomb gauge is more fundamental.

10. Why Coulomb gauge works cleanly

The lecture then explains how to reach Coulomb gauge using the gauge freedom: if $\nabla\cdot \vec A \neq 0$, choose a gauge function $f$ such that the transformed field satisfies

$$\nabla\cdot \vec A' = 0.$$

It also explains that fixing Coulomb gauge essentially exhausts the gauge freedom, leaving only time-independent residual freedom with

$$\nabla^2 f = 0,$$

which is then fixed by initial conditions and can be tolerated within ordinary Hamiltonian dynamics. :contentReference[oaicite:11]{index=11}

So Coulomb gauge removes the arbitrary gauge evolution and leaves only the genuine dynamical electromagnetic degrees of freedom.

11. Transverse polarization basis in Fourier space

To impose

$$\nabla\cdot \vec A = 0,$$

the lecture moves to Fourier space. It introduces polarization vectors $\vec\epsilon_j(\vec k)$ orthogonal to $\vec k$:

$$\vec\epsilon_j(\vec k)\cdot \vec k = 0, \qquad |\vec\epsilon_j|=1, \qquad \vec\epsilon_1\cdot \vec\epsilon_2 = 0.$$

These are the transverse polarization vectors. The lecture notes that this is analogous to using a basis of linearly polarized electromagnetic modes, and it also introduces the circular polarization basis

$$\vec\epsilon_\pm = \frac{1}{\sqrt2}(\vec\epsilon_1 \pm i\vec\epsilon_2),$$

which it says is more convenient because its conjugation properties are simpler. From this point on it uses circular polarization notation. :contentReference[oaicite:12]{index=12}

The vector potential is then expanded as

$$\vec A(\vec x,t) = \frac{1}{(2\pi)^{3/2}} \sum_{\pm}\int d^3k\, Q_\pm(\vec k,t)\,\vec\epsilon_\pm(\vec k)\,e^{i\vec k\cdot \vec x},$$

which automatically satisfies Coulomb gauge. :contentReference[oaicite:13]{index=13}

This is the key technical step that leaves only the two transverse electromagnetic modes.

12. Eliminating $A_0$ in Coulomb gauge

In Coulomb gauge, the lecture substitutes

$$A_0(\vec x,t) = -\frac{1}{4\pi}\int d^3x'\, \frac{j^0(\vec x',t)}{|\vec x' - \vec x|},$$

so only the instantaneous Coulomb solution remains. The harmonic gauge freedom has already been fixed away by the gauge choice. The lecture also defines the Fourier transform of the charge density $j^0(\vec x,t)$ and rewrites $A_0$ in momentum space as a $1/k^2$ term. :contentReference[oaicite:14]{index=14}

This is the origin of the instantaneous Coulomb interaction term in the Coulomb-gauge Hamiltonian.

13. The Coulomb-gauge Lagrangian in mode space

Substituting the transverse-mode expansion and the Coulomb expression for $A_0$ into the electromagnetic Lagrangian, the lecture rewrites the gauge-field Lagrangian entirely in terms of the transverse mode amplitudes $Q_\pm(\vec k,t)$ and the source current. After using the reality conditions and the circular polarization basis, it obtains a mode-space Lagrangian involving:

• the kinetic term $|\dot Q_\pm|^2$,
• the wave term $c^2k^2 |Q_\pm|^2$,
• the coupling of the transverse modes to the transverse current,
• and the Coulomb term proportional to

$$\frac{|j^0(\vec k)|^2}{k^2}.$$

The lecture pays attention to some factors of 2 coming from the reality condition and the circular polarization convention. :contentReference[oaicite:15]{index=15}

This is the Lagrangian already reduced to the true electromagnetic dynamical degrees of freedom.

14. Canonical momenta and Hamiltonian in Coulomb gauge

From the reduced Lagrangian, the lecture computes the canonical momenta of the mode amplitudes:

$$P_\pm(\vec k) = \frac{\partial L}{\partial \dot Q_\pm(\vec k)}.$$

Then it performs the Legendre transform to obtain the electromagnetic Hamiltonian in Coulomb gauge:

$$H_A = \int d^3k\, \left[ \frac12 \sum_{\pm} \left( \frac{|P_\pm(\vec k)|^2}{\hbar c} + \hbar c\,k^2 |Q_\pm(\vec k)|^2 \right) - \hbar c \sum_{\pm} Q_\pm(\vec k)\,\vec\epsilon_\pm^\ast(\vec k)\cdot \vec J^{\,T}(\vec k) + \frac{\hbar c}{2}\frac{|J^0(\vec k)|^2}{k^2} \right],$$

up to the lecture’s normalization conventions. The boxed result on the lecture slide is exactly the Hamiltonian it wants to use from this point onward. :contentReference[oaicite:16]{index=16}

This is the Hamiltonian form of Maxwell’s equations in Coulomb gauge.

15. Physical meaning of the Coulomb-gauge Hamiltonian

The lecture then explains what this Hamiltonian means physically.

It says that although the formula may look unfamiliar, the good old electromagnetic fields are still there. It writes them in terms of the mode amplitudes:

• the electric field contains a transverse dynamical part plus an instantaneous Coulomb part from the charge density,
• the magnetic field is purely transverse.

The slide explicitly notes that the Coulomb piece has no retardation, which is why this is called Coulomb gauge. The transverse parts are excited by the current density and can support waves at the speed of light. :contentReference[oaicite:17]{index=17}

This is a very good conceptual summary:

• the Coulomb field is instantaneous,
• radiation is carried by the transverse dynamical modes.

That is exactly how electrodynamics is split in Coulomb gauge.

16. What if we had chosen another gauge?

The lecture then asks a natural question: would a different gauge choice have produced a different Hamiltonian?

The answer is yes.

For example, in Lorentz gauge the first two free-field terms would be the same, but the coupling terms to the sources would look different. Within Lorentz gauges there are even sub-options, such as advanced or retarded Green’s functions instead of the instantaneous Coulomb Green’s function. But the lecture stresses that all these possible Maxwell Hamiltonians are canonically equivalent to the Coulomb-gauge Hamiltonian. They differ only by canonical transformations. :contentReference[oaicite:18]{index=18}

So the gauge-fixed Hamiltonian is not unique, but the underlying Hamiltonian field theory is.

17. No single Hamiltonian is gauge invariant

The lecture then makes an important conceptual point.

The unique Maxwell–Dirac Lagrangian field theory is gauge invariant and Lorentz invariant. But no single Hamiltonian is gauge invariant all by itself. A gauge transformation changes the form of the Hamiltonian, albeit to a canonically equivalent one. Likewise, no single Maxwell Hamiltonian is Lorentz invariant in any simple explicit sense. A Lorentz transformation changes the values of the fields and therefore changes the numerical value of the Hamiltonian, which is energy after all. :contentReference[oaicite:19]{index=19}

So the right picture is:

• one unique gauge-invariant Lorentz-invariant Lagrangian theory,
• many gauge-fixed Hamiltonian realizations,
• all canonically equivalent.

That is the actual structure of Hamiltonian gauge theory.

18. Why gauge invariance still matters in QFT

The lecture then asks: if any one canonically equivalent Hamiltonian is enough to define the quantum theory, does gauge invariance still matter?

Its answer is emphatically yes.

There are infinitely many possible Maxwell–Dirac Hamiltonians, but all of them lie inside a much smaller class of Hamiltonian field theories with very special properties. Those special properties come from the fact that they arise from a gauge-invariant Lagrangian. The lecture says that these properties are critical in quantum field theory because they ensure renormalizability. It specifically mentions the Ward–Takahashi identities as the crucial special feature of quantum gauge theories, while also noting that these are hard to prove directly in the Hamiltonian framework. :contentReference[oaicite:20]{index=20}

So gauge invariance is not a cosmetic symmetry. It is what makes the theory workable at the quantum level.

19. Why the course will later switch to path integrals

The lecture ends with a candid statement: it is one of the first major technical difficulties of QFT that the powerful simplifying symmetries of Lorentz and gauge invariance are not explicitly present in the canonical formulation. After using the Hamiltonian formalism to construct QED in the first place, the course will eventually move to a Lagrangian/path-integral formulation in order to restore Lorentz and gauge invariance explicitly. That will simplify calculations enormously. But first, one has to define the quantum theory at all — and for that, the Hamiltonian formalism is what the course is using now. :contentReference[oaicite:21]{index=21}

This is the real endpoint of the lecture: Hamiltonian gauge theory is necessary, but it is not the final computational language of QED.

Worked Examples

Example 1: Why $A_0$ is not dynamical

If the electromagnetic Lagrangian contains no $\dot A_0$, then

$$\Pi_0 = \frac{\partial \mathcal L}{\partial \dot A_0} = 0.$$

This is a universal constraint, independent of the initial conditions. So $A_0$ cannot be treated like an ordinary canonical field coordinate with its own independent momentum. Instead it is fixed by a constraint equation up to gauge freedom.

Example 2: Why only two photon polarizations remain

In Coulomb gauge,

$$\nabla\cdot \vec A = 0,$$

so in Fourier space the vector potential must be orthogonal to $\vec k$. Therefore only two independent polarization vectors remain for each $\vec k$. Expanding in those transverse polarization vectors leaves exactly two canonical coordinate-momentum pairs for the electromagnetic field. Those are the two physical photon polarizations.

Intuition

The main intuition of this lecture is that gauge symmetry removes fake dynamics.

At first the electromagnetic 4-potential $A_\mu$ seems to contain four field degrees of freedom. But two of them are not true dynamical variables in the Hamiltonian sense. One part is constrained universally, and another part is completely arbitrary because it is pure gauge. After those are removed, only the two transverse propagating degrees of freedom remain.

Coulomb gauge makes this especially visible:

• the longitudinal and scalar pieces are not propagated as independent waves,
• the scalar part becomes the instantaneous Coulomb potential,
• the transverse pieces are the actual radiative electromagnetic field.

So the Hamiltonian formalism strips the gauge field down to its real dynamical content.

Common Mistakes

• Thinking a local symmetry only means “one free function” and forgetting that it affects two phase-space coordinates per spatial point.
• Forgetting that local symmetry produces both constraints and arbitrary gauge freedom.
• Treating $A_0$ as an ordinary dynamical field with an independent conjugate momentum.
• Missing why $\Pi_0 = 0$ is a universal constraint.
• Thinking gauge fixing is optional if one wants a Hamiltonian formulation. It is not.
• Confusing the instantaneous Coulomb field with the transverse radiative field.
• Thinking a different gauge would define a different physical theory rather than a canonically equivalent Hamiltonian description.
• Expecting any one gauge-fixed Hamiltonian to be manifestly gauge invariant or Lorentz invariant.
• Underestimating why gauge invariance still matters once a gauge has been fixed.
• Missing the point that Hamiltonian QFT is a construction tool, while path integrals later become the symmetry-friendly computational tool.

Short Summary

The lecture begins from the gauge-invariant Maxwell–Dirac Lagrangian and asks for a Hamiltonian formulation of electrodynamics. It first develops the general Hamiltonian meaning of local symmetry: a local one-parameter symmetry affects two phase-space coordinates per spatial point, producing both universal constraints and arbitrary gauge freedoms. Applied to the electromagnetic field, this means that the naive eight phase-space variables associated with $A_\mu$ and its canonical momenta are reduced by four. The first universal constraint is

$$\Pi_0 = 0,$$

since the Lagrangian contains no $\dot A_0$. The second comes from the Euler–Lagrange equation for $A_0$, which fixes its Laplacian in terms of the charge density and the divergence of $\dot{\vec A}$, leaving only a harmonic gauge freedom. Since the remaining gauge freedom is arbitrary, a practical Hamiltonian formulation requires gauge fixing. The lecture chooses Coulomb gauge,

$$\nabla\cdot \vec A = 0,$$

and expands $\vec A$ in transverse polarization vectors in Fourier space, leaving only the two transverse dynamical electromagnetic modes. The scalar potential $A_0$ is eliminated in favor of the instantaneous Coulomb integral over the charge density. The resulting Coulomb-gauge Maxwell Hamiltonian contains:

• free transverse electromagnetic oscillator terms,
• coupling of the transverse modes to the transverse current,
• and an instantaneous Coulomb term proportional to $|J^0(\vec k)|^2/k^2$. This Hamiltonian correctly reproduces Maxwell’s equations: the electric field consists of transverse radiation plus the instantaneous Coulomb field, while the magnetic field is purely transverse. The lecture then emphasizes that other gauge choices would yield different but canonically equivalent Hamiltonians. No individual Hamiltonian is itself manifestly gauge invariant or Lorentz invariant, even though the underlying Maxwell–Dirac Lagrangian field theory is both. Finally, the lecture stresses that gauge invariance remains crucial in QFT because the special structures it implies, such as Ward–Takahashi identities, ensure renormalizability. This is also why the course will later move from Hamiltonian QED to the Lagrangian/path-integral formulation, where Lorentz and gauge invariance become manifest again. :contentReference[oaicite:22]{index=22}

Practice Problems

1. Why does a one-parameter local symmetry remove more dynamical structure than a one-parameter global symmetry?

2. Explain why $A_0$ does not behave as an ordinary dynamical variable in Hamiltonian electrodynamics.

3. What is the physical meaning of the universal constraint $\Pi_0=0$?

4. Why does the $A_0$ Euler–Lagrange equation act as a constraint rather than a dynamical evolution equation?

5. Why is gauge fixing necessary before writing a practical Hamiltonian for electrodynamics?

6. What is the difference between axial gauge, Lorentz gauge, and Coulomb gauge in the lecture’s discussion?

7. Why does Coulomb gauge leave only two transverse electromagnetic degrees of freedom?

8. What is the physical meaning of the instantaneous Coulomb term in the Coulomb-gauge Hamiltonian?

9. Why are different gauge-fixed Hamiltonians physically equivalent?

10. Why is no single gauge-fixed Hamiltonian manifestly gauge invariant?

11. Why does gauge invariance still matter after gauge fixing, especially in quantum field theory?

12. Why does the lecture say that path integrals will later become necessary even though the Hamiltonian formalism is being used now?