# Distinct choice of partition in the Path Integral

Practically all books in Quantum Mechanics and Quantum Field Theory define the non-relativistic path integral by taking one interval $$[a,b]$$ and breaking it up into $$N$$ subintervals of equal length.

Precisely this means one picks the partition $$P_N = \{t_0,\dots, t_N\}$$,

$$a = t_0 < t_1 < \cdots < t_{N-1} < t_N = b$$

where $$t_k = t_0 + k\epsilon\quad \epsilon = \dfrac{b-a}{N}.$$

Thus, with respect to this $$N$$-subintervals partition one discretizes the functional $$\mathfrak{F}$$ of interest - usually $$\mathfrak{F}[x(t)]=e^{iS[x(t)]}$$ or its euclidean counterpart $$\mathfrak{F}[x(t)]=e^{-S_E[x(t)]}$$.

Thus the path integral gets approximated by $$N+1$$ integrals over the reals:

$$\int \mathcal{D}x(t) \mathfrak{F}[x(t)]\sim \int_{-\infty}^ \infty\cdots\int_{-\infty}^\infty \mathfrak{F}_N(x_0,\dots, x_N)dx_0\cdots dx_N,$$

and the $$N\to \infty$$ limit defines the path integral.

Now, why does one choose the equal length subintervals? It seems to me that if we had chosen a distinct sequence of partitions $$(P_N)$$ with norms $$|P_N|\to 0$$ as $$N\to \infty$$ the result would be different.

Just let me make my point clear: one wants to discretize the paths and so wants to break the interval into a number of subintervals, and continue breaking those subintervals so that the lenght of these get to zero as the number of subintervals get to infinity.

There are infinitely many ways to do that, but physicists pick one specifically: the one on which the subintervals always are uniformly distributed in length.

I see no reason why the result shouldn't depend on this choice. Actually I've tried discussing this dependence on the choice of time slicing here on Math.SE and it seems that (1) the result does depend on the choice and (2) the definition which makes the result not depend on the choice would render the functionals of interest in Physics non-integrable.

So, if the results depend on the choice of ways to slice the interval $$[a,b]$$, why do Physicsts pick this one? Is there some discussion of this in the physics literature?

Or am I wrong and the results do not depend on the way the interval is sliced?

• It is well understood since L Cohen 1972 that some results depend on recipes of splicing and choice of intercalation, yielding different quantum ordering prescriptions out of "classical" inputs, when there are quantization ambiguities. In fact, there is a groaning cottage industry discussing such issues ad infinitum. For most velocity - independent potentials, however, such details don't matter, as handled in turgid path integration books, and physicists are given a pass to shrug, mumble, and move on to do the simplest thing. – Cosmas Zachos Jan 2 at 0:51
• Required reading, then, is Zinn-Justin, Grosche & Steiner, Langouche et al. – Cosmas Zachos Jan 2 at 0:58
• Dowker 1975 is an easy entry point to the (wan and dismal) field. – Cosmas Zachos Jan 2 at 1:12

Perhaps paradoxically, in much of quantum field theory these choices are not important. This is because the process of taking the continuum limit in a field theory involves fine tuning the couplings so that the the system goes through a second-order phase transition in which the correlation length becomes very long compared to the underlying lattice. It is by now well understood that in the neighbourhood of such phase transitions the exact details of the system have no effect. All that matters are the symmetries, and the space-time dimension. This is "critical point universality." The basic idea (originating in experiments on critical phenomena, and explained theoretically by Leo Kadanoff and Ken Wilson) is a higher-dimensional analogue of the central limit theorem from statistics: any experiment measures only averages over some resolution-defined volume, and - as with the division by $$\sqrt{N}$$ in the central limit theorem- after rescaling ("wavefunction renormalization") the details are washed out, and the measured field values are drawn from a "renormalization group fixed point" distribution.