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2 votes

Lorentz-invariant phase space integral

Using $$\frac{1}{2E}=\int\! dp^0 \,\theta(p^0)\, \delta(p^2-m^2),$$ one obtains $$\begin{align}J &: =\int\! \frac{d^3p_3}{2 E_3} \frac{d^3p_4}{2E_4} \, \delta^{(4)}(p_1+p_2-p_3-p_4) \delta(t-(p_1-...
Hyperon's user avatar
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2 votes

Some integrals in QED Renormalisation

Your suspicion is justified. You are indeed missing a crucial approximation not mentioned in your post, turning a simple few-line calculation into an exercise in self-torture. The photon mass $m_\...
Hyperon's user avatar
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1 vote

Some integrals in QED Renormalisation

Your complaint is not entirely justified, the first integrand is just a rational function, for which solution methods can easily be found: integration-rational-functions Although tedious, it can be ...
Jos Bergervoet's user avatar
2 votes
Accepted

How to derive the magnetic field at a distance $x$ (in the same plane) from the centre of a circular loop with current $I$ and radius $R$?

Your formula is not really a derivative, it's more of differential element (of an integral). Moreover, you have cast the vector nature of the problem to the wind: $$ d\vec B(\vec r) = \frac{\mu_0}{4\...
JEB's user avatar
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2 votes

Proving a Grassmann integral identity

Using the hint in @Qmechanic's answer I am able to prove the identity Method 1: $$ \begin{align} & \frac{\partial}{\partial \eta_{1}}\frac{\partial}{\partial \bar\eta_{1}} \left(1+a \left(\bar{\...
Faber Bosch's user avatar
4 votes
Accepted

Proving a Grassmann integral identity

Hints: Method 1: Use that Berezin integration is the same as differentiation, and that the Taylor series for the exponentials truncate. Method 2: Use the WKB/saddle point/stationary phase ...
Qmechanic's user avatar
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3 votes

What is the homogeneous charge density (vs inhomogeneous)?

A charge density is something that you integrate over space to get a total charge. This means that you have to be careful what you mean when you write down its functional form—the total integrated ...
Rokas Veitas's user avatar
1 vote

Hooke's law and derivation of work done by spring confusion

Let us consider the most generic case: say your equilibrium position is $x_{\text{eq}}$ and that you want to compute the work done by stretching/compressing from position $x_1$ to position $x_2$. We ...
Aschkal's user avatar
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1 vote

Hooke's law and derivation of work done by spring confusion

The force required from some already stretched/compressed, has the same form. In other words, $F$ is also $kx$, in this condition. $x_i$ is not just the initial point, it is the equilibrium point.
heon's user avatar
  • 101
1 vote

How to know the position of an object when calculating the center of mass, without using integrals?

Try to balance the 1/4 piece of pizza with one finger.
Hyperon's user avatar
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6 votes
Accepted

Exponential decay of propagator outside lightcone

Such asymptotic behaviour are typically calculated using the Laplace method (which is generalised to the saddle point method). It's worth looking into in depth, you'll use it again and again in QFT. ...
LPZ's user avatar
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0 votes

Electric field at a point created by a charged object (derivation/integration process)

It is often the case that the electric field is found by integrating Coulomb's law: $$\vec E(\vec r)=k{q\over |\vec r-\vec r^\prime|^3}(\vec r-\vec r^\prime).$$ As you note in your post, we can write ...
Albertus Magnus's user avatar
0 votes

Integral Quantity Interpretation

Apply the standard expansion of the vector identity, see $$\nabla (\mathbf a \cdot \mathbf b) = (\mathbf a \cdot \nabla)\mathbf b + \mathbf a \times (\nabla \times \mathbf b) + (\mathbf b \cdot \...
hyportnex's user avatar
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