Ok, so I looked into it a bit, and here’s my understanding:

The Polyakov action is kinda by default manifestly Lorentz invariant, but in order to quantize it, one generally first picks the light cone gauge, where this gauge choice treats some of the coordinates differently, losing the manifest Lorentz invariance. The reason for making this gauge choice is in order to make unitarity clear (/sorta automatic).

An alternative route keeps manifest Lorentz invariance, but proceeding this way, unitarity is not clear.

And then, in the critical dimensions (26 or 10, as appropriate; We have fermions, so, presumably 10) it can be shown that a certain issue (chiral anomaly, I think it was) gets cancelled out, and therefore the two approaches agree.

But, I guess, if one imposes the light cone gauge, if not in a space of dimensionality the critical dimension, the issue doesn’t cancel out and Lorentz invariance is violated? (Previously I was under the impression that when the dimensionality is wrong, things just diverged, and I’m not particularly confident about the “actually it implies violations of Lorentz invariance” thing I just read.)

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ekjhgkejhgk a day ago | parent [-]

> losing the manifest Lorentz invariance.

You understand that this have nothing to do with actual Lorentz invariance, yes? It sounds like you don't really understand the meaning of those terms you're using.

Do you understand what "manifest Lorentz invariance" means?

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drdeca a day ago | parent [-]

Yes? It means the Lorentz invariance is automatic from the form of the expression, does it not?

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ekjhgkejhgk a day ago | parent [-]

Yes. But when "Lorentz invariance isn't automatic from the form of the expression" it does NOT follow that you don't have Lorentz invariance.

	▲	drdeca 5 hours ago \| parent [-]
		Of course. Did part of what I said suggest I thought otherwise? I guess the part about the “when you quantize it after fixing the gauge in a way that loses the manifestness of the Lorentz invariance, if you aren’t in the critical dimension, supposedly you don’t keep the Lorentz invariance” part could imply otherwise? If that part is wrong, my mistake, I shouldn’t have trusted the source I was reading for that part. I was viewing that part as being part of how you could be right about Lorentz invariance being something derived nontrivially from the theory. Because, the Polyakov action (and the Nambu-Goto action) are, AIUI, typically initially(at the start of the definition of the theory) formulated in a way that is not just Lorentz invariant, but manifestly Lorentz invariant, and if there is no step in the process of defining the theory that isn’t manifestly Lorentz invariant, I would think that Lorentz invariance wouldn’t be a nontrivial implication, but something baked into the definition throughout, so, for it to be a nontrivial implication of the theory, at some point after the definition of the classical action, something has to be done that, while it doesn’t break Lorentz invariance, it “could” do so, in the sense that showing that it doesn’t is non-trivial. And, I was thinking this would start with the choice of gauge making it no longer manifestly Lorentz invariant. I trust you have much more knowledge of string theory than I do, so I would appreciate any correction you might have.