## Main Question or Discussion Point

I read in an article that a quantum field is one where every point in the field is defined by an imaginary number. If you square the imaginary number you get a wave function. But can a three dimensional field be defined by a set of points, finite or infinite? Does it mean a field characterized by an array of points? What kind of an array would characterize a field? How resolved would it have to be?

Related Quantum Physics News on Phys.org
Avodyne
I read in an article that a quantum field is one where every point in the field is defined by an imaginary number.
The article was wrong.

A quantum field is an operator (or set of operators) at every point in space.

bhobba
Mentor
I read in an article that a quantum field is one where every point in the field is defined by an imaginary number.
Avodyne is correct. A quantum field is defined by an operator at each point. The meaning of the operator is as per an observable in QM:
http://physics.mq.edu.au/~jcresser/Phys301/Chapters/Chapter13.pdf

Thanks
Bill

Nugatory
Mentor
I read in an article that a quantum field is....
It's impossible for us to comment sensibly when you don't provide a pointer to the article. Either it's wrong (likely, especially if it's a source that wouldn't be allowed under the PhysicsForums rules), or you've misunderstood it, but we can't tell which.

The article was wrong.

A quantum field is an operator (or set of operators) at every point in space.
Thank you so much for responding. However, I can't get past the statement, "everywhere in space." It makes absolutely no sense, logically or mathematically. You can describe space in all kinds of ways, in many different systems, but you cannot account for a field as just a set of points. I'm sure its just something I can't get my head around, but also, what do they mean by an imaginary number that if squared gives the wave function.

bhobba
Mentor
Thank you so much for responding. However, I can't get past the statement, "everywhere in space." It makes absolutely no sense, logically or mathematically. You can describe space in all kinds of ways, in many different systems, but you cannot account for a field as just a set of points.
A field is not a set of points - its something assigned to each point in a set of points.

Mathematically it's a function whose domain are the points of space, and as such can be defined with complete rigour using set theory.

An electric field you almost certainly learnt about at school is the most common example.

Did you express similar doubts to your teacher at the time :p:p:p:p:p:p:p:p

Just kidding of course - these concepts can be slightly tricky at first - but persevere.

Thanks
Bill

Last edited:
bhobba
Mentor
I'm sure its just something I can't get my head around, but also, what do they mean by an imaginary number that if squared gives the wave function.
What that is probably referring to is the Born rule:
http://en.wikipedia.org/wiki/Born_rule

What you wrote is incorrect.

Thanks
Bill

Last edited:
A field is not a set of points - its something assigned to each point in a set of points.

Mathematically it's a function whose domain are the points of space, and as such can be defined with complete rigour using set theory.

An electric field you almost certainly learnt about at school is the most common example.

Did you express similar doubts to your teacher at the time :p:p:p:p:p:p:p:p

Just kidding of course - these concepts can be slightly tricky at first - but persevere.

Thanks
Bill

Did you express similar doubts to your teacher at the time :p:p:p:p:p:p:p:p Bill, I shall ignore this rather gratuitous remark. But are the points of space little three dimensional balls? two dimension circles? are there an infinite number of whatever they are? a finite number? can space curve depending on our perspective? is space continuous in all magnitudes i.e. very small magnitudes. Maybe space is limited by a "smallest possible length." A field may be a characterization of space, a mental projection, an idea, a way of understanding space, or orienting ourselves and physical processes. So a physical phenomenon can define space as a spatial relation, and information can be expressed in terms of that spatial relation, but there is no way in this universe or any other that space is composed of an infinite set of points.

Drakkith
Staff Emeritus
But are the points of space little three dimensional balls? two dimension circles? are there an infinite number of whatever they are? a finite number?
I believe a point is, by definition, 0-dimensional. As such there should be an infinite amount of them.

So a physical phenomenon can define space as a spatial relation, and information can be expressed in terms of that spatial relation, but there is no way in this universe or any other that space is composed of an infinite set of points.
Points are not physical objects, but mathematical ones. If you take any region of space, you can assign coordinates to any point within that region, and mathematically there will be an infinite number of points available to assign a coordinate to.

• bhobba
bhobba
Mentor
But are the points of space little three dimensional balls? two dimension circles? are there an infinite number of whatever they are? a finite number? can space curve depending on our perspective? is space continuous in all magnitudes i.e. very small magnitudes. Maybe space is limited by a "smallest possible length." A field may be a characterization of space, a mental projection, an idea, a way of understanding space, or orienting ourselves and physical processes.
Think back to Euclidean geometry - what was the definition of a point?
http://aleph0.clarku.edu/~djoyce/java/elements/bookI/bookI.html
'A point is that which has no part.'

Since it has no part you can always find a point between any two points so there must be an infinite number.

Thanks
Bill

Last edited:
bhobba
Mentor
but there is no way in this universe or any other that space is composed of an infinite set of points.
Your reason for saying that is?

Physics is a mathematical model. The mathematical model par-excellence is good old Euclidean geometry:

Physics in general follows exactly the same paradigm.

Thanks
Bill

Last edited:
I believe a point is, by definition, 0-dimensional. As such there should be an infinite amount of them.

Points are not physical objects, but mathematical ones. If you take any region of space, you can assign coordinates to any point within that region, and mathematically there will be an infinite number of points available to assign a coordinate to.

A set with zero elements, or a set with an infinite number of elements of zero magnitude can exist in only one place, and that is the imagination. If you use a mathematical construct that exists in the imagination, as ideas let us say, and that construct proves to have a valid application in the physical world then as reasonable men we can assume a realm of mathematical reality apart from physical reality. So the question I am asking is, in which reality does space reside, the mathematical one or the real one. Let's say you decide to open a pharmacy. You ride through the Bronx, get the lay of the land, see, for example that there are three medical centers on Third Ave and no pharmacies. At this point the pharmacy exists only in your mind, as an idea, like the elements in your set that have no dimensions. To make the pharmacy a reality, you are going to need some energy, or money in this case, and I'm going to have to make a set of decisions based on a lot of things. If I do all that, then something which only existed in my mind has become a physical reality. But in perusing the area, I was gauging spatial relations. The "area" could sustain a pharmacy. Each decision I made crystallized the idea into a reality. And to my way of thinking the same principle applies to space. Space can be an approximation of a field, a very good approximation, but as a physical reality and an approximation it cannot be infinite. And I identify it as a physical reality for two reasons. First it is flat, and secondly it is limited to a smallest possible length. So as physical reality and informational system it is a finite set and limited with respect to the functions that can be applied to it.

I read in an article that a quantum field is one where every point in the field is defined by an imaginary number. If you square the imaginary number you get a wave function. But can a three dimensional field be defined by a set of points, finite or infinite? Does it mean a field characterized by an array of points? What kind of an array would characterize a field? How resolved would it have to be?
However, I can't get past the statement, "everywhere in space." It makes absolutely no sense, logically or mathematically.
The article was wrong.

A quantum field is an operator (or set of operators) at every point in space.
Hello everyone
Article was wrong - true
Quantum field is an operator (and it is already squared) - no need to square again. To get wave function you had to cube the velocity of particle for example photon - it is finite number. Than quantum field it is not two dimensional field anymore it is three dimensional wave.
In this case it is wrong to say "everywhere in space" - correct is "everywhere in time" according to STR where the mass had a energy in photon field.

http://phys.org/news/2014-12-quantum-physics-complicated.html#jCp Quantum physics just got less complicated

Drakkith
Staff Emeritus
Space can be an approximation of a field, a very good approximation, but as a physical reality and an approximation it cannot be infinite.
Uh, no, you don't know this and you have zero evidence to support your position.

And I identify it as a physical reality for two reasons. First it is flat, and secondly it is limited to a smallest possible length.
General Relativity disagrees with you.

Since it is obvious you are trying to force science to bend to your personal opinion, I am locking this thread.

• vanhees71, goce2014 and bhobba