26 votes

Why didn't the Klein-Gordon equation suggest antimatter like the Dirac equation did?

In the beginnings of quantum theory, people were looking at the K-G and the Dirac equation as equations for wave functions (or at least something similar that would give them a probability density ...
  • 111k
21 votes

Is Maxwell's field the wave function of the photon?

Weinberg is right. The issue here is with the usual interpretation of the wavefunction as an amplitude density. This implies being able to localize the particle in an arbitrarily small region. ...
  • 1,604
18 votes

Difference between field and wavefunction

Ignore spin, polarization, and even Lorentz issues, absorb all superfluous constants, and consider time and one space dimension. A one-particle complex wavefunction $\psi(x,t)=\langle x|\psi \...
18 votes

Does relativistic quantum mechanics (RQM) really violate causality?

The actual difference is in how these approaches treat measurements. In the single particle theory, your observable is the particle coordinates $x^i(t)$. Measuring them at $t_1$ and $t_2$ can lead to ...
16 votes

What's wrong with the square root version of the Klein-Gordon equation?

Secondly, why is the first differential equation cumbersome to work with? It seems like it would in fact be easier to work with, since the operator under the square root could be expanded in terms of ...
  • 2,665
16 votes
Accepted

Noether's Theorem and scale invariance

First of all, let's see what Noether's Theorem says about your specific case (Klein-Gordon under global rescaling of the fields). Noether's theorem states that To every differentiable symmetry of ...
14 votes

Is it true that the Schrödinger equation only applies to spin-1/2 particles?

As per Rob's suggestion, I decided to make this an answer. (Addendum: I've been meditating on this very topic for some time, and have been directed to some interesting literature referenced on ...
  • 2,665
14 votes
Accepted

A question on using Fourier decomposition to solve the Klein Gordon equation

Notation: $x=(t,\boldsymbol x)$; $k=(k_0,\boldsymbol k)$; $kx=k_0t-\boldsymbol k\cdot\boldsymbol x$; $\mathrm dx=\mathrm dt\;\mathrm d^3\boldsymbol x$; etc. You can in principle perform the Fourier ...
13 votes
Accepted

How does QFT interpret the Negative probability problem of the real scalar fields' Klein-Gordon equation?

Quantum field theory solves the problem by giving a different interpretation to the "probability". In the case of complex fields, quantum field theory also introduces antiparticles. In the first-...
12 votes
Accepted

The Lagrangian in Scalar Field Theory

I have a slightly different perspective from the other two answers which provides a more elementary motivation. Suppose you know nothing about renormalizability or energy-momentum relations and all ...
  • 1,494
12 votes
Accepted

Why is the Klein Gordon equation of second order in time?

There's a major difference between Schrödinger/Dirac equations and Klein-Gordon one: the former are complex while the latter is real. But if you think of them a little, you'll also find a major ...
  • 26.9k
11 votes
Accepted

Bessel function representation of spacelike KG propagator

This can be seen by partial integration $$\frac{\partial}{\partial \rho}\sqrt{\rho^2-m^2}=\frac{\rho}{\sqrt{\rho^2-m^2}}$$ OP edit: More explicitly, we use this to write $(3)$ as \begin{align} D(...
10 votes
Accepted

Contradictory result for scalar-field propagator from Feynman rules and LSZ formula

My own attempt at this: the first result is wrong, and the second one is right but incomplete. Feynman: in the diagram, all the lines are external, so there is no propagator in the diagram. Therefore,...
10 votes
Accepted

Interacting Fields in QFT

You are always free to define $$a_{\boldsymbol k}\equiv \int\mathrm d^3 x\ \mathrm e^{ikx}(\omega_{\boldsymbol k}\phi(x)+i\pi(x)) \tag{1}$$ where $\pi=\dot\phi(x)$. If you take the time derivative of ...
10 votes

Why can the Klein-Gordon field be Fourier expanded in terms of ladder operators?

You might feel more comfortable if we run the reasoning 'in reverse' a bit. I'm basically just recounting what all the standard texts do, but without jumping ahead in the interpretation. We begin ...
  • 96k
10 votes
Accepted

Hamiltonian Field Theory in Peskin & Schroeder

When you transform from the Lagrangian to the Hamiltonian picture, you necessarily must choose a particular foliation of spacetime -- that is, you must single out a particular time direction, and ...
  • 2,404
9 votes
Accepted

How to derive the theory of quantum mechanics from quantum field theory?

The field and the wavefunction look similar, but they don't really have much to do with each other. The main point of the field is to group the creation and annihilation operators in a convenient way, ...
  • 26.1k
9 votes

Quantizing a complex Klein-Gordon Field: Why are there two types of excitations?

There is a conceptually simple (but fiddly) way of relating this expansion to the usual Fourier expansion. TL;DR: Requiring $\phi$ to satisfy the Klein-Gordon equation divides the nonzero Fourier ...
9 votes

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 ...
  • 26.1k
9 votes

Does relativistic quantum mechanics (RQM) really violate causality?

OP has a point. On one hand, P&S on p. 14 argue that in first quantized RQM in the operator formalism the propagator is $$\begin{align}&\langle {\bf x}_f,\tau_f \mid {\bf x}_i,\tau_i\rangle\cr ...
  • 174k
8 votes

The Lagrangian in Scalar Field Theory

A reasonable motivation for that Lagrangian can be found making a classical analogy. The kinetic energy in classical physics is proportional to the square of the rate of change of the position with ...
  • 308
8 votes

Is the Dirac equation equivalent to the Klein-Gordon equation for its left handed component?

The Dirac equation is more restrictive than the Klein-Gordon equation. For every solution to the Dirac equation, its components will be a solution of the Klein-Gordon equation, but the converse isn't ...
  • 5,474
8 votes

Can't the Negative Probabilities of Klein-Gordon Equation be Avoided?

I'll work in the $+---$ convention so $\mathbf{k}\cdot\mathbf{x}-k_0t=-k\cdot x$. You should find that $\psi=\exp{\mp ikx}$ obtains $j^\mu=\pm m^{-1}k^\mu$. Complex conjugation of $\psi$ still gives a ...
  • 22.8k
8 votes
Accepted

Time-independent Klein-Gordon PDE

The "time-independent" Schrodinger equation is called so because it doesn't contain time derivatives. The physical solutions, however, do contain explicit time dependence, as the energy eigenstates ...
  • 8,084
7 votes
Accepted

breitenlohner freedman stability condition

For a quick (and somewhat dirty) way of deriving the bound, do the following: Recall, that in Poincaré coordinates, the metric of $AdS_{d+1}$ is $$ ds=\frac{1}{z^2}\left(dz^2+\eta^{\mu\nu}dx^\mu dx^\...
  • 3,392
7 votes

Klein-Gordon inner product

The Klein-Gordon inner product is a natural construction for functions defined on the mass hyperboloid $k^2=m^2$, because if you write your function in momentum space, $$ \phi(x)\sim \int\widetilde{\...
7 votes

What is $\phi(x)|0\rangle$?

The quantum mechanical interpretation in terms of probabilities of being at a point in space is intrinsically nonrelativistic. To get this interpretation for a relativistic particle, one needs to ...
7 votes
Accepted

Quantizing a complex Klein-Gordon Field: Why are there two types of excitations?

The point is that the quantization procedure is usually only valid for real-valued physical observables. All versions of treat the classical observables as real functions on phase space (things get ...
  • 111k
7 votes
Accepted

The Klein-Gordon field

Scalar means $$ U(\Lambda)\phi(x) U(\Lambda)^\dagger=\phi(\Lambda x) $$ or, in other words, it transforms trivially under the Lorentz Group, as opposed to, say, vectors or spinors, which transform as $...

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