Why is it so hard to rigorously construct interacting QFTs…
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Why is it so hard to rigorously construct interacting QFTs in 4d spacetime?
Having concluded my bachelor's in physics I'm transitioning towards mathematical physics for my master's and one of the first questions that made me realize I'm actually quite interested in the field of mathematical physics is this one (besides of course that I'm fascinated by rigorous and unambiguous approaches to a field that sits so close to my heart).
I know Lorentz invariance in itself causes a whole lot of issues, the big one that immediately comes to my mind is covariant quantization of gauge theories: fix a gauge that is not lorentz invariant and the naive approach to canonical quantization works just fine, but add this constraint back and suddendly you get a physicist to ramble about negative norms in a Hilbert space (the slander comes from a place of love, I like teasing my physics dept friends).
But these kinds of problems seem, to some extent, secondary. In fact, with some clever "work arounds" one can formally solve these issues, whereas the more fundamental task of simply \_defining\_ an interacting theory in 3+1 dimensions seems to be still out of reach. Why? I still know very little about constructive QFT but as far as I understand it interacting QFTs in lower dimensions have successfully been defined, while in 3+1 dimensions we only have rigorous constructions of free theories, so I guess the problem isn't the presence of the interactions per se but it's specifically the number of dimensions?
#science
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Having concluded my bachelor's in physics I'm transitioning towards mathematical physics for my master's and one of the first questions that made me realize I'm actually quite interested in the field of mathematical physics is this one (besides of course that I'm fascinated by rigorous and unambiguous approaches to a field that sits so close to my heart).
I know Lorentz invariance in itself causes a whole lot of issues, the big one that immediately comes to my mind is covariant quantization of gauge theories: fix a gauge that is not lorentz invariant and the naive approach to canonical quantization works just fine, but add this constraint back and suddendly you get a physicist to ramble about negative norms in a Hilbert space (the slander comes from a place of love, I like teasing my physics dept friends).
But these kinds of problems seem, to some extent, secondary. In fact, with some clever "work arounds" one can formally solve these issues, whereas the more fundamental task of simply \_defining\_ an interacting theory in 3+1 dimensions seems to be still out of reach. Why? I still know very little about constructive QFT but as far as I understand it interacting QFTs in lower dimensions have successfully been defined, while in 3+1 dimensions we only have rigorous constructions of free theories, so I guess the problem isn't the presence of the interactions per se but it's specifically the number of dimensions?
#science
source
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