# Tag Info

Accepted

### How does canonical quantization work with Grassmann variables?

When you are dealing with Grassmann numbers you have a "left derivative" and a "right" derivative. A left derivative removes the variable from the left, a right derivative removes it from the right. ...
• 1,644

### Are all fields in the universe we know of quantum fields?

Currently all fundamental fields are quantum, except for gravity. For this reason Quantum Gravity is a hot area of research, but the full Quantum Gravity theory has not been developed yet. Why not? ...
• 12.6k

### Why can't $p^0$ change sign under a proper orthochronous Lorentz transformation?

Sometimes, diagrams are more useful than equations! Note that $p^2 = - (p^0)^2 + |\vec{p}|^2$ remains unchanged under any Lorentz transformation. Suppose that $p^2 < 0$ (say $p^2 = -1$). A plot of ...
• 25.6k

### Does the Newtonian gravitational field have momentum analogous to the Poynting vector?

No, in Newtonian mechanics there is no gravitational momentum. Physically, the reason is that the gravitational field doesn't propagate: it responds instantaneously to changes in the matter ...
• 28k
Accepted

### Can Einstein-Hilbert action be derived from symmetry considerations?

Well, one can reason as follows: One wants a diffeomorphism-invariant action, which must be of the form $$S=\int d^4x\sqrt{-g}L,$$ where $L$ is a scalar in terms of transformation properties. ...
• 10.8k
Accepted

### When is Schwartz's method for "integrating out" a field valid?

I find that Schwartz is using the laziest way to discuss the integration of a field. What is going on can be made much more transparent rather easily. To simplify the discussion, I will assume that ...
• 11.8k

### Invariance of Action vs. Lagrangian in Noether's theorem?

No, they are not the same. To see why, even in classical mechanics, suppose we have symmetry transformation $q \rightarrow q + \epsilon K$ that leaves the Lagrangian invariant. This means that we must ...
Accepted

### What is "gradient energy" in classical field theory?

The best intuition is from the example of a "loaded string": a set of $N$ masses that are connected by a massless elastic string. If we imagine the limit as $N \to \infty$ but with the ...
• 48.3k

### Why do we need to make a tensor for the electromagnetic field?

The main reason why one needs a (2,0)-tensor is the fact that neither the electrical field $\mathbf{E}$ (3d-vector) nor the magnetic field $\mathbf{B}$ (3d-vector) can be fitted into a 4-vector, i.e. ...
• 9,438
Accepted

### Counting Degrees of Freedom in Field Theories

You really should split your question. I will answer the part where you do not understand how counting of degrees of freedom work. Basically we count the number of propagating (physical) degrees of ...
• 2,520
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### What is the actual form of Noether current in field theory?

Eq. (5) is (up to factors of the infinitesimal parameter $\varepsilon$) the standard expression for the full Noether current. Here: $\delta x^{\mu}$ is the so-called horizontal component of the ...
• 200k

### Klein gordon field and positive/negative energy solutions

You say you're doing classical field theory, but the terminology comes from QM: these terms are only positive and negative energy if you interpret the field $\phi$ as a wavefunction, as people did ...
• 28k
Accepted

### Classical field limit of the electron quantum field

The classical limit of bosonic quantum mechanical systems with both finite and infinite degrees of freedom is pretty well understood from a mathematical standpoint (with complete rigour, and for quite ...
• 11.9k

### Why is action a functional of $q$ only?

The notation $q$ in the functional $S[q]$ stands for the whole parametrized curve/path $q:[t_i,t_f]\to \mathbb{R}$, not just a single position. In particular, the parametrized path already carries all ...
• 200k
Accepted

Accepted

### Is the Four-gradient of a Scalar Field a Four-Vector?

The $4$-gradient is a $4$- vector. Formally, when $x^\mu\to x'^\mu=\Lambda^\mu{}_\nu x^\nu$  \begin{align*} \partial'_\mu &=\frac{\partial}{\partial x'^\mu}\\ &=\frac{\partial}{\partial (\...
• 2,668
Accepted

### Gauge symmetry of massive vector field

Symmetries of the action must be considered without use of the equations of motion. An on-shell symmetry is a vacuous notion - if you use the equations of motion, as you do when using eq. (5) to ...
• 124k
Accepted

### Schroedinger equation for wave functional (QFT)

Recall, from Hatfield's textbook (QFT of point particles and strings) & Jackiw's review that the functional equation you are emulating is just that, a functional equation the extension of an ...
• 61.9k