Quantum Field Theory: completing the “A Bridge between Mathematicians and Physicists” series

I decided to read the series “A Bridge between Mathematicians and Physicists” written by Eberhard Zeidler. But when I read the preface of the first book I realized that at first this series should be composed of six volumes, namely:

  1. Quantum Field Theory I: Basics in Mathematics and Physics
  2. Quantum Field Theory II: Quantum Electrodynamics
  3. Quantum Field Theory III: Gauge Theory
  4. Quantum Field Theory IV: Quantum Mathematics
  5. Quantum Field Theory V: The Physics of the Standard Model
  6. Quantum Field Theory VI: Quantum Gravity and String Theory

I have only the first three books. Searching for the last three I discovered that Eberhard Zeidler died and therefore the last three volumes will never be published.

For this reason I ask: could someone please indicate me books that cover the subject of the last three books and that focus a lot on the mathematical part?

Thank you for your attention.

In the Physicists’ definition of the path integral, does the result depend on the choice of partitions?

The usual definition of the path integral in QM usually goes as follows:

  1. Let $ [a,b]$ be one interval. Let $ (P_n)$ be the sequence of partitions of $ [a,b]$ given by $ $ P_n=\{t_0,\dots,t_n\}$ $ with $ t_k = t_0 + k\epsilon$ where $ \epsilon = (b-a)/n$ , and $ t_0 = a$ , $ t_n=b$ .

  2. Let $ \mathfrak{F}: C_0([a,b];\mathbb{R}^d)\to \mathbb{C}$ be a functional defined on the space of continuous paths on $ [a,b]$ . One defines its discretization as the set of functions $ \mathfrak{F}_n : \mathbb{R}^{(n+1)d}\to \mathbb{C}$ given by $ $ \mathfrak{F}_n(x_0,\dots,x_n)=\mathfrak{F}[\xi_n(t)]$ $ where $ \xi_n(t)$ is the curve defined by taking the partition $ P_n$ , defining $ \xi(t_i)=x_i$ and linearly interpolating between the points – in other words $ \xi(t)$ is for $ t\in [t_i,t_{i+1}]$ the straight line joining $ x_i$ and $ x_{i+1}$ .

  3. One defines the functional integral as the limit $ $ \int_{C_0([a,b];\mathbb{R}^d)}\mathfrak{F}[x(t)]\mathcal{D}x(t)=\lim_{n\to \infty}\int_{\mathbb{R}^{(n+1)d}} \mathfrak{F}_n(x_0,\dots, x_n) d^dx_0\dots d^dx_n$ $

    if it exists.

  4. In the case of interest for physics one has $ \mathfrak{F}[x(t)]=e^{iS[x(t)]}$ or rather $ \mathfrak{F}_E[x(t)]=e^{-iS_E[x(t)])}$ the euclidean version.

So by slicing the time axis into equal subintervals, one converts the functional to a sequence of functions, integrates those and takes the limit.

This is the construction outlined for instance in Peskin’s book or Sakurai’s book, just rewritten in a more “mathematical” form.

Now, if on the very first step we choose another sequence of partitions $ (P_n)$ such that the sequence of partition’s norms $ |P_n|\to 0$ as $ n\to \infty$ but which is not the sequence of equal subintervals, would the resulting path integral be different?

I don’t see reason why it should be equal. The intervals endpoints are distinct, the interpolations are distinct, hence the maps $ \mathfrak{F}_n$ are distinct.

If it is I think this is a big problem. After all, the way we are slicing the time axis is arbitrary and just one trick to make the problem easier to deal with.