Wave function of particle / quantum field in space, also in time?

[Silvio Macedo]

TL;DR Summary

Quantum fields have wave functions that determine a particle position in space. It solves non-locality, double-slit paradox, tunnel effect, etc. What if the wave function is also in time? Won't it solve the breaking of causality at quantum level? (I did search before posting)

Quantum fields have wave functions that determine a particle position in space. It solves non-locality, double-slit paradox, tunnel effect, etc. What if the wave function is also in time? Won't it solve the breaking of causality at quantum level? (Delayed Choice/Quantum Eraser/Time)

Not much else to say. I'm a mere engineer with less than basic knowledge of quantum theories. But it just occurred to me that if in addition to probability in space, as defined by the wave function of a particle, that same wave function somehow spreads the existence of a particle in time, it would for example solve the the delayed choice paradigm too.

Until collapsed, the wave function "exists" in space and time; that is, it projects the probability of the particle being both space and time; when the wave function is collapsed, by an observer, it collapses in a classically-coherent fashion both in space and time.

And like non-locality, and tunnel-effect, this does not cause causality to break; it simply allows for a certain quantum fuzziness until we force the universe to settle down.

Apologies if I'm just wasting your time.
Any input - even if it is just a link for me to figure it out by myself - will be appreciated.
(I did search before posting)
Thank you!

 

[Nugatory]

that is, it projects the probability of the particle being both space and time; when the wave function is collapsed, by an observer, it collapses in a classically-coherent fashion both in space and time.

There are two misunderstandings here, one about what the theory says and one about what the math says.

First, about the theory: An observer does not collapse the wave function. This idea was part of early thinking about quantum mechanics but was abandoned in later decades as the theory was properly formalized. A good non-technical overview of why the theory no longer needs an observer in the naive sense of Schrodinger's cat and popular explanations of the double-slit experiment can be found in David Lindley's book "Where does the weirdness go?".

Second, about the math: The post-collapse wave function is just as much a wave function as the pre-collapse wave function. For example, if I start with a particle in a superposition of spin-up and spin-down and send it through a spin-measuring device, it will collapse into either the state spin-up or spin-down - but those states are no more classical than the original state, they're superpositions of spin-left and spin-right. A good starting for understanding what collapse is really doing might be Giancarlo Ghirardi's "Sneaking a look at god's cards".

I should add that neither of these books will be a substitute for a real textbook and a semester-long class... but they're a good introduction for people for whom that is not an option.