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I am trying to make sense of statements about unitarity in this popular science article about Nima and Jaroslav's new idea.

My first query is that it is claimed that unitarity is a pillar of quantum field theory:

Locality and unitarity are the central pillars of quantum field theory

In second quantization, however, I recall nothing of unitarity, and one can construct quantum field theories that are not unitary e.g., in the Standard Model without a Higgs boson, the $WW\to WW$ elastic scattering matrix element is greater than unity for $\sqrt{s}\gtrsim1 \textrm{TeV}$. Is it correct that unitarity is a pillar of quantum field theory?

My second query regards a described experiment for measuring unitarity. It is suggested that one repeatedly measures the final state of a scattering process:

To prove [unitarity], one would have to observe the same interaction over and over and count the frequencies of the different outcomes. Doing this to perfect accuracy would require an infinite number of observations using an infinitely large measuring apparatus, but the latter would again cause gravitational collapse into a black hole.

Take, e.g., a simplified experiment in which one repeatedly measures the state $$ \psi = a \phi +b\chi. $$

What would the results for repeated measurements of the state be for $|a|^2+|b|^2>1$? and $|a|^2+|b|^2<1$?

Upon measurement, I would be entangled with the state. I could only see one outcome per measurement. Maybe an omniscient god could see that states were destroyed for $|a|^2+|b|^2<1$ and created for $|a|^2+|b|^2>1$, if he saw all worlds in a many worlds interpretation. But I would see nothing odd? So how is this experiment supposed to work?

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  • $\begingroup$ What do you mean by "exceed unitarity" ? $\endgroup$ – Siva Sep 20 '13 at 7:31
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    $\begingroup$ If unitary is not true, the optical theorem will not be true, and there are consequence about inequalities that cross-sections verify. So, by checking cross-sections, you may see possible violations of unitarity. $\endgroup$ – Trimok Sep 20 '13 at 8:28
  • $\begingroup$ @Siva I've made some edits to make it clearer. $\endgroup$ – innisfree Sep 20 '13 at 8:49
  • $\begingroup$ @Trimok that is a good point, but how does it relate to a simpler setup with quantum mechanics that I described? Maybe I have misunderstood the proposed experiment - if all they mean is a precision test of cross sections, it's a funny way to say it. $\endgroup$ – innisfree Sep 20 '13 at 8:55
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Nima and Jaroslav aren't really planning to build any non-unitary theories, at least not at this point. Their results for the amplitudes are exactly the same as they are in the normal treatment, so they're also unitary. Unitarity seems to be necessary for consistency, as you seem to agree.

What the claim is that their machinery for calculating the scattering amplitudes doesn't make any assumptions that would make locality manifest. In the operator treatment, unitarity is manifest from the Hermiticity of the Hamiltonian that generates the evolution; in the path integral, it really follows from the reality of the action in the path integral.

In Nima's and Jaroslav's picture, unitarity is a derived geometric feature of their polytopes with their volume forms. Similar comments hold not only for unitarity but also locality. In that case, it makes sense to believe that their formalism will allow one to construct new physically consistent theories that will not be local (in the sense of locality of QFTs), perhaps new formulations of quantum gravity. In the normal "fields in spacetime" approach, nonlocal or non-unitary theories may be "immediately identified". But in the amplituhedron framework, it's hard to pinpoint which theories of a similar kind are unitary and/or local so one is more naturally led to new generalizations (at least in the case of locality).

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  • $\begingroup$ Thanks, that certainly clears up some aspects of the article, and also the point at which unitarity comes into QFTs. Can you explain the (thought) experiment for proving unitarity? i.e. could one measure anything odd in the non-unitary $<1$ and $>1$ cases? or is the exp't just a precision test of cross sections? $\endgroup$ – innisfree Sep 20 '13 at 8:59
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"To prove [unitarity], one would have to observe the same interaction over and over and count the frequencies of the different outcomes. Doing this to perfect accuracy would require an infinite number of observations using an infinitely large measuring apparatus, but the latter would again cause gravitational collapse into a black hole."

For the infinite number of observations, without even considering scattering, imagine a photon with a specific polarization $|\psi\rangle = \alpha |0\rangle+ \beta |1\rangle$. Suppose you want to "measure" this polarization, how are you going to proceed ?

First, you will choose axis corresponding to $|0\rangle$ / $|1\rangle$ or $\frac{1}{\sqrt{2}} (|0\rangle + |1\rangle)$ / $|\frac{1}{\sqrt{2}}(0\rangle - |1\rangle)$.

Clearly, if you have only one photon, you will make one measurement on one axis, this will give you "statistically" some information, but you will be unable to extract $\alpha$ and $\beta$. In fact, even with a finite number of experiments, with a finite number or photons in the same state, you will have "statistically" more information about $\alpha$ and $\beta$, that is :you will have "statistically" better and better precision about them, but you will be unable to extract $\alpha$ and $\beta$.

"statistically" is a mean. In fact, you cannot exclude, that, for instance,
doing one billion experiments with measurement in the basis $|0\rangle$ / $|1\rangle$, you will find one billion time the state $|0\rangle$ after measurement.

So, you need to make an infinite number of experiments, with an infinite number of photons in the same state, and varying the axis of measurement, and only this will allow you to know perfectly $\alpha$ and $\beta$.

So, in scattering, to prove unitarity, you have to check exactly all the scattering amplitudes, and this needs an infinite number of observations.

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  • $\begingroup$ That certainly makes sense, but it seems to be making a mountain out of a mole-hill. For example, is there an identical problem in (classical) statistics? Suppose I wanted to know whether a coin was fair, I would have to flip it an infinite number of times to be sure? In fact, you require an infinite number of experiments if you want any measurement (classical or quantum) with absolute precision... $\endgroup$ – innisfree Sep 20 '13 at 9:56
  • $\begingroup$ Is there something I am not getting here? Is there some deeper point about being able to perform measurements in principle in quantum mechanics that is relevant? e.g. interpretations of QM, postitivism etc? $\endgroup$ – innisfree Sep 20 '13 at 10:00
  • $\begingroup$ @innisfree : Yes, in any probabilistic system, classical or quantum, you will have the same problem. Checking unitarity means checking an exact equality, so you have to check exactly probabilities/cross-sections, so you have to do a infinite number of measurements. $\endgroup$ – Trimok Sep 20 '13 at 10:05
  • $\begingroup$ So what really has this experiment got to do with unitarity? I could make my own thought experiment, To prove a coin is fair, one would have to flip it over and over and count the frequencies of different outcomes. Doing this to perfect accuracy would require an infinite number of flips etc $\endgroup$ – innisfree Sep 20 '13 at 10:11
  • $\begingroup$ @innisfree : As I said in my previous comment, "Checking unitarity means checking an exact equality, so you have to check exactly probabilities/cross-sections" $\endgroup$ – Trimok Sep 20 '13 at 10:14

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