3 ms·
The list could go on ... if you are a mathematician. If you are a physicist (not working on string theory), you rarely use topological concepts - if at all. W
by aroberge 14y ago
The list could go on ... if you are a mathematician. If you are a physicist (not working on string theory), you rarely use topological concepts - if at all. While working towards my Ph.D. on some topic in finite temperature effects in gauge theories, I did not know anything about fiber bundles and still know nothing about de Rham cohomology classes. My non-theorist physicists friends knew even less about those mathematical topics.
- xyzzyz 14y agoSure. That's why math is great: after you have spent enough time familiarizing with concepts, you can easily understand so many things in terms of concepts you already feel very comfortable with. You see electromagnetism, realize that electromagnetic field is just 2-form on 4-manifold, and see that Maxwell equations just state that both this form and its Hodge dual are closed, so that they represent de Rham cohomology classes. Of course, you can also represent this result in a classical way, but creating a bridge translating physical concepts into well studied mathematical frameworks has the advantage enabling you to also pass the bridge in the other direction: sometimes you can find physical interpretation concepts that arised in the abstract setting. So, we know that forms representing electromagnetic fields are closed, but what does it mean in physics when they are exact? How to interpret Mayer-Vietoris sequence in electromagnetism terms? What the induced maps do with forms representing electromagnetic fields, and how to interpret homotopy? What about Poincare duality? It's such a great feeling to realize that your favourite toy, after playing with it for years, is actually able to do a whole lot more stuff than you were previously aware of.