Spinless Salpeter equation non-locality

• ChrisVer
In summary, the conversation discusses the process of generalizing the Schrodinger equation to a relativistic equation, resulting in the Spinless Salpeter equation. It is mentioned that the Hamiltonian in this case is non-local, which can be seen by looking at the propagator or the Lagrangian. This is because the kinetic energy operator is expanded to an infinite series, with the momentum operator being a derivative operator that differentiates to infinite order, making the theory affected by points far from the one being considered.
ChrisVer
Gold Member
A straightforward attempt to generalize the Schrodinger equation to a relativistic equation, is obtained by taking the Eigenvector/value equation:
$H |a_k> = E_k |a_k>$

with a hamiltonian of the form $H= T+V$,

In the relativistic case we have $T=\sqrt{p^2+m^2}$ , and taking also (for my purposes) the case that $V(\vec{x})=0$ we get the Spinless Salpeter equation:

$\sqrt{p^2 +m^2} |a_k> = E_k |a_k>$

(http://arxiv.org/abs/hep-ph/9807342 , below eq.1)

I was wondering, how can someone see immediately that this form of the Hamiltonian is non-local? I think that someone sees this by looking at the propagator? Or does someone get this by writting the Lagrangian and seeing it's non-local (contains terms that are not functions of the fields or their derivatives)?

Last edited:
The point is that you can taylor expand the kinetic energy operator w.r.t. p and it will be an infinite series. But the momentum operator in that series is a derivative operator and that means that operator(T) is in fact differentiating to infinite order. But the higher the derivative, the more points you need and those points get farther from the point your considering which, for an infinite order differentiation, means the points very very far from a point affect the properties of that point and so the theory is non-local.

1. What is the Spinless Salpeter equation?

The Spinless Salpeter equation is a non-relativistic quantum mechanical equation that describes the behavior of a single particle in a non-local potential. It takes into account both the spin and the mass of the particle and is commonly used in atomic, nuclear, and particle physics.

2. What does non-locality mean in the Spinless Salpeter equation?

Non-locality in the Spinless Salpeter equation refers to the fact that the potential energy term depends on the positions of both particles involved in an interaction, rather than just the position of the particle being described. This allows for a more accurate description of interactions between particles that are not in close proximity.

3. How does the Spinless Salpeter equation differ from other equations in quantum mechanics?

The Spinless Salpeter equation differs from other equations in quantum mechanics, such as the Schrödinger equation or the Klein-Gordon equation, in that it takes into account the spin of the particle. This makes it particularly useful in studying particles with non-zero spin, such as atoms or nuclei.

4. What is the significance of the Spinless Salpeter equation in physics?

The Spinless Salpeter equation has played a significant role in the development of modern physics, particularly in the fields of atomic, nuclear, and particle physics. Its ability to accurately describe the behavior of particles with spin has led to important discoveries and advancements in these areas.

5. How is the Spinless Salpeter equation solved?

The Spinless Salpeter equation is typically solved using numerical methods, such as the variational method or the perturbation method. These methods involve approximating the solution to the equation through a series of calculations, often aided by computer programs.

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