I’m derelict in mentioning this but ElKement’s blog, Theory And Practice Of Trying To Combine Just Anything, has published the second part of a non-equation-based description of quantum field theory. This one, titled “May The Force Field Be With You: Primer on Quantum Mechanics and Why We Need Quantum Field Theory”, is about introducing the idea of a field, and a bit of how they can be understood in quantum mechanics terms.

A field, in this context, means some quantity that’s got a defined value for every point in space and time that you’re studying. As ElKement notes, the temperature is probably the most familiar to people. I’d imagine that’s partly because it’s relatively easy to feel the temperature change as one goes about one’s business — after all, gravity is also a field, but almost none of us feel it appreciably change — and because weather maps make the changes of that in space and in time available in attractive pictures.

The thing the field contains can be just about anything. The temperature would be just a plain old number, or as mathematicians would have it a “scalar”. But you can also have fields that describe stuff like the pull of gravity, which is a certain amount of pull and pointing, for us, toward the center of the earth. You can also have fields that describe, for example, how quickly and in what direction the water within a river is flowing. These strengths-and-directions are called “vectors” [1], and a field of vectors offers a lot of interesting mathematics and useful physics. You can also plunge into more exotic mathematical constructs, but you don’t have to. And you don’t need to understand any of this to read ElKement’s more robust introduction to all this.

[1] The independent student newspaper for the New Jersey Institute of Technology is named The Vector, and has as motto “With Magnitude and Direction Since 1924”. I don’t know if other tech schools have newspapers which use a similar joke.

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## Author: Joseph Nebus

I was born 198 years to the day after Johnny Appleseed. The differences between us do not end there. He/him.
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Thanks again for your kind pingback and publicity :-)

I need to get to vectors and tensors in the next post(s) but I am still trying to figure out how to do this without mentioning those terms. Fluid dynamics is often a good starting point, e.g. to introduce, ‘derive’ or better motivate Schrödinger’s equation. On the other hand Feynman used to plunge directly into path integrals – presenting them as a rule along the lines of “This is the way nature works – live with it” – and deriving Schrödinger’s equation later.

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I’m not quite sure how I’d do either. I think I could probably explain vectors without having to use mathematical symbolism, since the idea of stuff moving at particular speeds in directions can call on physical intuition. Tensors I don’t know how I’d try to explain in popular terms, partly because I’m not really as proficient in them as I should be. I probably need to think seriously about my own understanding of them.

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I have also always considered easier to imagine the different aspects of a vector – the abstract object and the ‘arrow’ as it lives in a specific base. But how do you really imagine the ‘abstract tensor object’ – in contrast to a ‘matrix’ (with more than 3 dimensions probably…)

I have started to read about general relativity (… will finish after I have finally understood the Higgs…) and it took me quite a while to comprehend that you are not allowed to shift a vector in curved space as you shift the ‘arrow’. Actually it made me think about vectors in a new way…

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(I’m embarrassed that I lost this comment somehow.)

I can sort of reconstruct the process when I think I started to get vectors as a concept, particularly in thinking of them as not tied to some particular point, or even containing information about a point, but somehow floating freely off that. If I get around to trying to explain vectors I might even be able to make all that explicit again.

Tensors I keep feeling like I’m on the verge of having that intuitive leap to where I have some mental model for how they work but I keep finding I don’t quite do enough work with them that it gets past following the rules and into really understanding the rules.

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