Schrodinger equation from Klein-Gordon?

Question

One can view QM as a 1+0 dimensional QFT, fields are only depending on time and so are only called operators, and I know a way to derive Schrodinger's equation from Klein-Gordon's one.

Assuming a field $\Phi$ with a low energy $ E \approx m $ with $m$ the mass of the particle, by defining $\phi$ such as $\Phi(x,t) = e^{-imt}\phi(x,t)$ and developing the equation

$$(\partial^2 + m^2)\Phi~=~0$$

neglecting the $\partial_t^2 \phi$ then one finds the familiar Schodinger equation:

$$i\partial_t\phi~=~-\frac{\Delta}{2m}\phi.$$

Still, I am not fully satisfied about the transition field $\rightarrow$ wave function, even if we suppose that the number of particle is fixed, and the field know acts on a finite dimensional Hilbert Space (a subpart of the complete first Fock Space for a specific number of particles). Does someone has a reference to another proposition/argument for this derivation?

Edit: for reference, the previous calculations are taken from Zee's book, QFT in a Nutshell, first page in Chapter III.5.

This post imported from StackExchange Physics at 2014-04-01 16:22 (UCT), posted by SE-user toot

Comments

David Tong does have a good derivation/explanation for this in here: damtp.cam.ac.uk/user/tong/qft/two.pdf I have however had my own doubts about it, in case you have the same doubts, here are threads that answer them: physicsforums.com/showthread.php?t=709980 physics.stackexchange.com/questions/77290/…

This post imported from StackExchange Physics at 2014-04-01 16:22 (UCT), posted by SE-user guillefix

Answer 1

I think you are mixing up two different things. Namely:

1. On the one hand, you can see QM as 0+1 (one temporal dimension) QFT, where the position operators (and their conjugate momenta) in the Heisenberg picture plays the role of the fields (and their conjugate momenta) in the QFT. You can check, for instance, that spatial rotational symmetry in the quantum mechanical theory is translated to an internal symmetry in the QFT.

2. On the other hand, you can take the non-relativistic limit (by the way, ugly name because Galilean relativity is as relativistic as Special relativity) of the Klein-Gordon or Dirac theory to get the Schrödinger QFT, where $\phi$ (in your notation) is a quantum field instead of a wave function. There is a chapter in Srednicki's book where this issue is raised in a simple and nice way. There, you can also read about spin-stastistic and the wave function of multi-particle states. Let me add some equations that hopefully clarify (I'm using your notation and of course can be wrong factors, units, etc.):

The quantum field is: $$\phi \sim \int d^3p \, a_p e^{-i(p^2/(2m) \cdot t - p \cdot x)}$$

The Hamiltonian is:

$$H \sim i\int d^3x \left( \phi^{\dagger}\partial_t \phi - (1/2m)\partial _i \phi ^{\dagger} \partial ^i \phi \right) \, \sim \int d^3p \, p^2/(2m) \,a^{\dagger}_p a_p$$

The evolution of the quantum field is given by:

$$i\partial _t \phi \sim [\phi, H] \sim -\nabla ^2 \phi /(2m)$$

1-particle states are given by:

$$|1p> \sim \int d^3p \, \tilde f(t,p) \, a^{\dagger}_p \, |0> $$

(one can analogously define multi-particle states)

This state verifies the Schrödinger equation:

$$H \, |1p>=i\partial _t \, |1p>$$ iff

$$i\partial _t \, f(t,x) \sim -\nabla ^2 f(t,x) /(2m)$$

where $f(t,x)$ is the spatial Fourier transformed of $\tilde f (t,p)$.

$f(t,x)$ is a wave function, while $\phi (t, x)$ is a quantum field.

This is the free theory, one can add interaction in a similar way.

This post imported from StackExchange Physics at 2014-04-01 16:22 (UCT), posted by SE-user drake

Comments

Thanks for the update, but I am specifically under a limit operation that would lead me to a "first quantization scheme" from a "second quantization scheme", alias, is it enough from the fact that I recover a Schrodinger equation and can then construct a conserved probability current with a positive density component ($j^0$) to reinterpret the field as a wave function whose modulus square gives probabilities?

This post imported from StackExchange Physics at 2014-04-01 16:22 (UCT), posted by SE-user toot

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I'm not sure what you are looking for. $f$ is a wave function that verifies the Schr. equation. The expectation value of $\phi$ is also a function that verifies the Schr. equation. So, as long as you can normalize them you get a quantum mechanical probabilistic interpretation. Have I answered your question?

This post imported from StackExchange Physics at 2014-04-01 16:22 (UCT), posted by SE-user drake

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I think you did =) Thanks you.

This post imported from StackExchange Physics at 2014-04-01 16:22 (UCT), posted by SE-user toot

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@drake: The expectation value of $\phi$ is not the correct way to extract wavefunction from field. The right way is to consider the state $\int \psi(x)\phi^\dagger(x)$ where $\psi$ is a number and $\phi^\dagger(x)$ is the nonrelativistic field.

This post imported from StackExchange Physics at 2014-04-01 16:22 (UCT), posted by SE-user Ron Maimon

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@toot: The comment above is the correspondence between nonrelativistic fields and wavefunctions. If you smear a nonrelativistic creation field with a function, you produce a particle with wavefunction $\psi$.

This post imported from StackExchange Physics at 2014-04-01 16:22 (UCT), posted by SE-user Ron Maimon

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Thanks. You are just changing my notation: your $\psi$ is my $f$. The expectation value is a solution of the equation, it is usually called the classical field.

This post imported from StackExchange Physics at 2014-04-01 16:22 (UCT), posted by SE-user drake