Could probability amplitude for a path equal a complex number whose length is always 1 and whose angle is the action divided by Planck's constant?

Question

I'm reading "Zee A. - Quantum Field Theory, as Simply as Possible", where near beginning of explanation of QFT he gives what appears to be path integral formulation, he states:

The corresponding probability equals the absolute value squared of the probability amplitude, that is, the square of the length of that complex number. (p.89)

probability amplitude for a path equals the complex number with length = 1 and angle $θ = S(path)/ħ$. (p.91)

we are instructed to sum up all these probability amplitudes to determine the probability amplitude to get from the starting point to the ending point. (p.92)

So, whereas in classical physics, the action for each path is a real number, a point on an infinitely long line, in quantum physics, the probability amplitude for each path is a complex number, represented by a point on a circle of radius 1. (p.91)

Wikipedia:

1. The probability for an event is given by the squared modulus of a complex number called the "probability amplitude".
2. The probability amplitude is given by adding together the contributions of all paths in configuration space.
3. The contribution of a path is proportional to $e^{iS/ħ}$

So A. Zee states probability amplitude vector has length of 1, whereas wikipedia states "proportionality". From A. Zee statement probability for a path will be 1 (and probability to get from the starting point to the ending point would be larger than 100%), which seems strange. Could it still be considered correct? How?

Answer 1

Consider a particle travelling from $x_i$ to $x_f$. Quantum mechanically, we could ask about the probability to have such process $$ p(x_i\rightarrow x_f) = |\langle x_f|\Psi(t)\rangle|^2 = |\langle x_f |U(t,t_0)|x_i\rangle|^2 $$ Both Wikipedia and Zee says

Wiki: The probability for an event is given by the squared modulus of a complex number called the "probability amplitude".

Zee:The corresponding probability equals the absolute value squared of the probability amplitude, that is, the square of the length of that complex number.

Both are talking about the equation above. The mentioned amplitude is $\langle x_i|U(t,t_0)|x_f\rangle$. By path integral formalism we know that $$ \langle x_i|U(t,t_0)|x_f\rangle = \int \mathcal D x \quad e^{iS[x]/\hbar} $$ where $S$ is the classical action of the particle. If we interpret this astonishing big integral (which is a collection of multiple integrals, one for each point in the path) as a sum, and the term inside as a "probability of a path", we could say that the probability amplitude of the particle travels the path $x$ is proportional to $e^{iS/\hbar}$. The term proportional here is important, since we have a normalization factor inside $\mathcal D x$. Wiki is more careful, and Zee is sacrificing the rigor in the sake of intuition.

Zee: we are instructed to sum up all these probability amplitudes to determine the probability amplitude to get from the starting point to the ending point.

Look that Zee is not being clear if he is calling "these probability amplitudes" the quantity proportional to $e^{iS/\hbar}$ or just the exponential. But being formal, there is no sense to assume that a probability of any path being $1$, since the sum will give us something bigger than 1 as you noticed, so not resulting in a probability.

The probability amplitude of a path being proportional to a exponential of a phase is a way to interpret the path integral. Again Wiki and Zee agree in such interpretation, but Wiki is more formal and rigorous

Wiki: The contribution of a path is proportional to eiS/ħ

Zee:probability amplitude for a path equals the complex number with length = 1 and angle θ = S(path)/ħ

Look that Zee says something about length = 1. He is interpreting the complex exponential as a rotating vector in the complex plane. Since the term is $e^{iS/\hbar}$, the length is $1$ in this sense.

In resume, Zee chose to talk about unormalized probability distribution, so you are right to assume that it does not sum up to $100%$. It is of course a very non rigorous way to phrase things, but he seems to trying to build some intuition behind such complex object. To be more rigorous, an integral is not a sum in first place, since we can't sum what we can't count. But it you want to things to sum to $1$, remember the proportionality and the normalization factor.

Answer 2

1. Zee is oversimplifying while using Feynman's stopwatch hand/arrow picture of the path integral.

   The main point is that (over an unbounded position space $\mathbb{R}^d$) the probability distribution (coming from the path integral) is only relative, i.e. its normalization is unphysical.

2. For details, see e.g. my related Phys.SE answer here and links therein.