# Transition amplitude vs. transition probability

In quantum mechanics, a physical system corresponds to a Hilbert space $\mathscr{H}$. States correspond (not in a one-to-one way) to points in $\mathscr{H}$ and the physical postulate is that the "transition amplitude" (a complex number) from a state corresponding to $v$ into a state corresponding to $u$ is given by: $$\mathscr{A}(v\to u) = \frac{\langle v,u \rangle}{\sqrt{\langle v,v \rangle\langle u,u \rangle}}$$

Where the physical meaning of the transition amplitude is that if you take the squared absolute value of this complex number, you get the actual probability of the system going from the state corresponding to $v$ into the state corresponding to $u$.

My question is: is there any physical information contained in the phase of $\mathscr{A}(v\to u)$?

If yes:

1) What is this meaning and how do we measure it in the laboratory?

2) When working with projective Hilbert spaces, this information (AFAIK) has to be discarded, because the transition amplitude between two rays (a ray is defined as an equivalence class of $\mathscr{H}$: the equivalence class corresponding to $v\in\mathscr{H}-\{0\}$ is given by $\left[v\right]\equiv\{u\in\mathscr{H}|\exists\lambda\in\mathbb{C}:u=\lambda v\}$) is defined as (for example Bargmann 1964): $$\mathscr{A}(\left[v\right]\to \left[u\right]) = \left|\frac{\langle v,u \rangle}{\sqrt{\langle v,v \rangle\langle u,u \rangle}}\right|$$ and this seems like a sensible definition because otherwise, setting $$\mathscr{A}(\left[v\right]\to \left[u\right]) = \frac{\langle v,u \rangle}{\sqrt{\langle v,v \rangle\langle u,u \rangle}}$$ results in a not well-defined map $\mathscr{A}$.

Thus, if indeed this information is discarded when working with projective Hilbert spaces, why is it allowed to discard it if it contains physical information?

• It seems to me you have answered your own question (2. answers 1.) - since the physical space of states where every point corresponds to a unique state is the projective Hilbert space, the phase cannot contain information. – ACuriousMind May 30 '15 at 22:04
• The definition of "physical information" is a bit more complicated in quantum mechanics than in classical mechanics. The "information" that we measure on the real systems is not identical to the "information" that we need to calculate the time evolution of these systems in the theory. This has caused endless and completely fruitless philosophical debates about "reality". In practice we have to use phase information in calculations, but in the end we have to drop it when we transition to measurable quantities. That's not the same thing as discarding it completely. – CuriousOne May 30 '15 at 22:48
• @CuriousOne, in that case, how can one use only projective Hilbert spaces to describe a physical system, if one is missing some of the information? – PPR May 31 '15 at 0:09
• You still need Fock spaces for multi-particle problems and you have to extend the definitions to include the coupling to thermal baths, i.e. the Hilbert space plus thermodynamic averages (density matrix), but these are in addition to your question. For single particles and without a thermal environments you can calculate everything within this simple formalism, as far as I know, but the formalism can't answer your question with regards to the "physical reality" of phases any more than Newton's laws can tell you why there are space, time and mass, but both theories agree with observation. – CuriousOne May 31 '15 at 0:26

It sounds to me like you are asking for the difference between relative and absolute phase. It is true that there is never a meaningful absolute phase of any complete state of an isolated system, for the reason you mention-- multiplying that state by any complex number has no effect on any observable. Also note that since your approach to defining states does not require that they be normalized, it's not just the absolute phase, but also the absolute magnitude, that has no physical significance. In a geometric picture, all that physically matters is the direction in which the state "points."

However, an important rule about states is that you can add superpositions of them to get a new state. Neither the phase nor magnitude of the new state matters either, but what does matter is the relative phase and magnitude of the states you are superimposing to get the new state, because they will affect the "direction" of the superimposed state.

So that suggests pretty strongly to me that the concepts of phase and magnitude, in the context of the physical information they represent, only have meaning in regard to the adding of states, i.e., when generating superpositions. Hence, the meaning of phase and magnitude are inextricably connected to the meaning of a superposition, and that's how they connect to the concept of interference in the two-slit experiment and so on.

• So how does superposition work in the projective Hilbert space picture? – PPR Apr 23 '17 at 14:58
• The projective Hilbert space is like seeing the usual (normalized) Hilbert space from any distance and any angle. By that I mean, imagine a sphere of unit vectors of various different colors. Now imagine tilting your head, and zooming in closer so they look bigger. That's like multiplying them all by the same complex number, and you regard that as the same space. Superpositions are when you sum and renormalize, so you just have to do that before or after you tilt your head and zoom in, not one of each. – Ken G Apr 23 '17 at 16:47
• In other words, in the projective Hilbert space, superposition is something you do to the equivalence classes, not something you do to arbitrarily chosen states from the equivalence classes. If you want to select a pair of states that superimpose the same way as the equivalence classes to yield a state in the equivalence class, then you have to select those states appropriately. So the equivalence allows aU + bV = a*(cU) + b*(cV), but only if the c is the same in both terms. – Ken G Apr 23 '17 at 18:29

Well, for an individual measurement, for example in the double slit experiment with a single electron, the phases are what make the interference pattern in the accumulation, so in this sense they have measurable effects.

In the superposition of photons to make up the classical electromagnetic wave, the phases are also important. This experiment which shows interference of two independent laser beams relies on the superposition of the amplitudes and the phase information is important.