The discussion revolves around the nature of quantum fields and whether they can be defined by a finite or infinite set of points in three-dimensional space. Participants explore the mathematical and conceptual implications of defining fields, particularly in the context of quantum mechanics and the representation of points in space.
Participants express multiple competing views regarding the definition and nature of quantum fields and the representation of points in space. The discussion remains unresolved, with no consensus on whether space can be composed of an infinite set of points or how quantum fields should be conceptualized.
Participants highlight limitations in understanding the definitions and implications of quantum fields and points in space, indicating a dependence on mathematical definitions and the potential for misunderstanding based on differing interpretations.
The article was wrong.Duhoc said:I read in an article that a quantum field is one where every point in the field is defined by an imaginary number.
Duhoc said:I read in an article that a quantum field is one where every point in the field is defined by an imaginary number.
Duhoc said:I read in an article that a quantum field is...
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.Avodyne said:The article was wrong.
A quantum field is an operator (or set of operators) at every point in space.
Duhoc said: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.
Duhoc said: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.
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.bhobba said: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 learned 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
Duhoc said: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?
Duhoc said: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.
Duhoc said: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.
Duhoc said:but there is no way in this universe or any other that space is composed of an infinite set of points.
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.Drakkith said: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.
Duhoc said: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?
Duhoc said:However, I can't get past the statement, "everywhere in space." It makes absolutely no sense, logically or mathematically.
Avodyne said:The article was wrong.
A quantum field is an operator (or set of operators) at every point in space.
Duhoc said: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.