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The Fourier series has terms like $\sin \left(\frac {2\pi n t}P \right)$ After time $P$ all the terms will repeat, so the Fourier series can only represent functions that have period $P$. Many functions in nature are periodic, so these series can represent a lot of things we are interested in. The fixed period makes the various terms orthogonal, so ...

4

One very popular view (as espoused by Max Tegmark) is that (quoting count_to_10) : math works because the universe is based on math http://www.scientificamerican.com/article/is-the-universe-made-of-math-excerpt/ https://en.wikipedia.org/wiki/Mathematical_universe_hypothesis Such a view was common from the time of Pythagoras, through to Kepler and ...

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Both examples you ask about are for gapless systems, but in your edit you then bring up some notions related to the gapped systems, so let me address both. Moreover it will turn out that understanding the gapped case well (which is simpler) almost naturally implies the gapless case. Topological invariants for band insulators Suppose we have our Brillouin ...

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As you say, $\gamma_{\mu \nu}$ is a metric. Then, $\gamma_{\mu \nu}=\gamma_{ \nu \mu}$ and it makes no difference which one you write down. This already deals with one half of your discrepancy. Now for the trace: In the einstein summation notation, a the trace of $\gamma_{\mu \nu}$ is $\gamma_{\mu}^{\; \mu}$. The looking at your \frac{\partial (det \gamma)...

21

This is definitely not a dumb question. If we work in a (linear) Hilbert space, then our inner product $\langle \cdot,\cdot \rangle$ induces the usual natural flat metric (given by $d(\psi,\phi) = || \psi - \phi ||$). However, often we take the viewpoint that our states are elements of projective Hilbert space $\mathbb CP^n$. Then it is more natural to ...

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As commentators have indicated Hilbert space is a vector space. A manifold is a space with an atlas-chart construction with maps on overlapping regions that define connection coefficients and ultimately curvature. It is certainly possible to think of a finite dimensional complex vector space that is a locally flat region in an otherwise curved space. This ...

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Hilbert space is a complex normed vector space equipped with an inner product, where this inner product comes from the norm on the space (the norm of a vector on the hilbert space is the square root of the inner product of the vector with itself), but for a curved space like the minkowski space we will use a minkowski metric that differs from the usual ...

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Hilbert spaces are vectorspaces by definition. If you interpret a vector space as a manifold (which you can do) then it's a flat manifold.

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There was an FQXi essay competition on this subject in Spring 2015: "Trick or Truth: the Mysterious Connection Between Physics and Mathematics" Here is the home page with links to winners and other entries: http://fqxi.org/community/essay/winners/2015.1 The competition is meant to encourage an informal style, readable by non-experts, but with some ...

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language more general than math Sure, some of us call it English. I can teach and understand many lessons from physics without diving into the math. What's hard to do is discover or prove physics without math. "A body at rest tends to stay at rest. A body in motion tends to stay in motion." I can throw tons of math at that but the idea can be ...

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There were two achievements of early humans and early civilization that led to us being able to state and understand the environment we lived in. One was writing, and one was math. Math started with counting, then adding, i.e. counting and arithmetic. Counting was invented very very early in the homo family by many different groups. Then arithmetic and math ...

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