# Exploring the Relationship between Feynman's "Arrow" and Shroedinger's "Psi"

• eb227
In summary: Also, the Dirac equation is a 4-component vector equation, while the Schroedinger equation is a single complex equation. It's kind of hard to see how one is the same as the other. That is, until you notice that if you break the 4-component vector into two 2-component vectors, then the equation can be written as a set of two complex equations. And if you break it into 4 1-component vectors (scalars) then it can be written as 4 coupled real equations.So, in that sense, the Dirac equation is just a generalisation of the Schroedinger equation to 4 component wavefunctions. And it can be reduced to the Schroedinger equation
eb227
On page 50 he states " The probability of an event is equal to the square of the length of an arrow called the "probility amplitude".
What, if any, is the relationship between Feynman's "arrow" and Shroedinger's wave function "psi".

eb227 said:
On page 50 he states " The probability of an event is equal to the square of the length of an arrow called the "probility amplitude".
What, if any, is the relationship between Feynman's "arrow" and Shroedinger's wave function "psi".

It's the same. The Schroedinger wave function "psi" is a specific "coordinate representation" of the "arrow" Feynman talks about (in the basis of functions of position).

Psi is a complex number. In the Argand or polar representation of complex numbers they are vectors ("little arrows") in the geometric plane spanned by the vectors 1 = (1,0) and i = (0,1).

A complex number a + ib has the polar representation of a vector of magnitude or length m making an angle of theta with the positive real axis, with m = sqrt(a^2+b^2) and theta = arctan (b/a).

Then the probabllity generated by psi = (a + ib) is (a+ib)(a - iB) = (a^2+b^2), which is the square of the magnitude, or length, of the vector in the polar represntation.

Does that imply that Shroedinger's wave equation is the central mathematical equation of QED as well as Quantum Mechanics?

Last edited:
eb227 said:
Does that imply that Shroedinger's wave equation is the central mathematical equation of QED as well as Quantum Mechanics?

No. But, the fundamental ideas in Schroedinger's equation do get carried over.

However, Schroedinger's equation can't be used in QED because it is not valid for relativistic particles. Instead, you have to use the Dirac equation, which makes the mathematics much more complicated. (Dirac wavefunctions must be spinors, rather that scalars.)

Parlyne said:
...you have to use the Dirac equation, which makes the mathematics much more complicated. (Dirac wavefunctions must be spinors, rather that scalars.)

Plus Feynman's method becomes nonconvergent in Minkowski spacetime. A problem that is fixed by Wick rotation, at the cost of losing the pretty motivation. I don't know if Feynman mentions that in QED.

Parlyne said:
No. But, the fundamental ideas in Schroedinger's equation do get carried over.

However, Schroedinger's equation can't be used in QED because it is not valid for relativistic particles. Instead, you have to use the Dirac equation, which makes the mathematics much more complicated. (Dirac wavefunctions must be spinors, rather that scalars.)

Well, you can still use Schroedinger's equation in the relativistic version, in the sense:

hbar/i d/dt |psi> = H |psi>

if you place yourself in a specific reference frame.
Only, |psi> stands for the state vector, say, in Fock space.
But usually, in QFT, we write the Schroedinger equation in time-integrated form, which leads us to the Dyson recursion formula (from which the Feynman diagrams are derived in the canonical quantisation).

The Dirac equation is now the equation of the field operator (and not of the quantum state, or "wave field" or whatever).

It is in fact a very funny interplay, that originally, the Schroedinger equation was (for a single particle), seen as a kind of field equation for a field psi over space/time in a non-relativistic version, that Dirac tried to generalise it to a relativistic field equation, and that it then turned out that Dirac's "field equation" needed a "second quantisation". That is, we had to look upon the Dirac equation as a *classical* field equation now, and that we had to apply the formalism of quantum mechanics to this classical field, and that in this quantum theory of the quantization of a classical field, the Schroedinger equation turned up, again, as the time evolution equation.
So, after all, the Dirac equation is NOT the relativistic variant of the Schroedinger equation, though it was initially derived that way.

The Schroedinger equation is the time evolution equation in Hilbert space ; the Dirac equation is a classical field equation.

However, things are even more confusing ! It can now be seen that a quantized classical field is equivalent to the quantum description of a set of particles with variable number, obeying Bose or Fermi statistics. It is then a matter of taste to see a QFT as the quantum version of a classical field, with "emerging particles" ; or to see a QFT as the quantum version of a set of particles with a given statistics and variable number, and see the "field" emerging. They are equivalent.

vanesch said:
So, after all, the Dirac equation is NOT the relativistic variant of the Schroedinger equation, though it was initially derived that way.

That is an important point that many introductory sources omit. It took me a while to work that out for myself.

## What is the relationship between Feynman's "Arrow" and Schrodinger's "Psi"?

The relationship between Feynman's "Arrow" and Schrodinger's "Psi" is that they both refer to concepts within quantum mechanics. "Arrow" refers to the direction of time in quantum systems, while "Psi" refers to the wave function that describes the probability of finding a particle in a certain location.

## How do Feynman's "Arrow" and Schrodinger's "Psi" relate to each other?

Feynman's "Arrow" and Schrodinger's "Psi" are related because the direction of time in quantum systems is determined by the evolution of the wave function described by "Psi". In other words, the "Arrow" of time is determined by the behavior of the wave function.

## What is the significance of Feynman's "Arrow" and Schrodinger's "Psi" in quantum mechanics?

Feynman's "Arrow" and Schrodinger's "Psi" are both fundamental concepts in quantum mechanics. They help us understand the behavior of particles at the quantum level and how the evolution of wave functions dictates the direction of time in quantum systems.

## How do scientists use Feynman's "Arrow" and Schrodinger's "Psi" in their research?

Scientists use Feynman's "Arrow" and Schrodinger's "Psi" in their research by applying them to various quantum systems and experiments. They use these concepts to make predictions about the behavior of particles and to understand the underlying principles of quantum mechanics.

## What are some real-world applications of Feynman's "Arrow" and Schrodinger's "Psi"?

Feynman's "Arrow" and Schrodinger's "Psi" have many real-world applications, including in the development of quantum computers, quantum cryptography, and quantum teleportation. They are also used in fields such as chemistry, biology, and materials science to understand the behavior of particles and molecules at the quantum level.

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