- Online Resources on mathematical QFT and foundations:
Pflaum, M.J.,
*The Geometry of Classical
and Quantum Fields*, Lecture Notes.

The *FANCY Project* on Functional Analysis and Noncommutative Geometry.

The CRing Project on Commutative Algebra and related topics.

- Distribution Theory and Pseudodifferential Operators:

Brouder, Ch., Nguyen Viet Dang, and Hélein, F.,
*A smooth introduction to the wavefront set,*
Journal of Physics A: Mathematical and Theoretical **47** (2014).

Shubin, M.A., *Pseudodifferential Operators and Spectral Theory,* Second Edition, Springer, Berlin, Heidelberg, New York 2001.

Markus: *A classic on the topic.*

Trèves, F.,*Topological Vector Spaces, Distributions and Kernels,* Academic Press Inc., New York, 1967.

Markus: *This is a comprehensive monograph on the functional analytic foundations of distribution theory. It also contains
an exposition of nuclear spaces and the Schwartz kernel theorem.*

- Mathematics of Quantum Mechanics and Quantum Field Theory:

Folland, G.B., *Quantum Field Theory : A Tourist Guide for Mathematicians,* AMS (2008).

Manny: *Honest mathematician's overview of traditional QFT as commonly taught and
presented in typical QFT textbooks. I found this to be a very instructive read.*

Takhtajan, L.A., *Quantum Mechanics for Mathematicians,* AMS (2008).

Manny: *Advanced mathematical treatment, not self-contained.
Sometimes not as explanatory as I would have liked. I couldn't determine the general formulation of classical
graded (super) mechanics from the chapter on this topic.*

Borcherds, R.E. and Barnard, A., *Lectures on Quantum Field Theory,*
arXiv:0204014 (2002).

Leonard: *These are Borcherd's UC Berkeley notes on quantum field theory, which contain some details that his paper
* Renormalization and Quantum Field Theory *glosses over. À la Borcherds, Lagrangians are defined as the symmetrization of
the sheaf of derivatives of classical fields, Feynman measures are defined as linear functionals on a rank-1 free module over the symmetrization of
the algebra of compactly supported sections of the sheaf of local actions, and renormalizations are defined as automorphisms of the symmetrization of
the sheaf of local actions, where the automorphisms are of a sheaf of Hopf algebras. The group of renormalizations is shown to act freely and
transitively on the set of Feynman measures, so there is no canonical Feynman measure, but there is a canonical orbit of such measures
under renormalization.*

- Axiomatic Quantum Field Theory:
Streater, R.F. and Wightman, A.S., * PCT, Spin, Statistics, and All That,* Basic Books.

Manny: *A classic mathematical introduction to axiomatic QFT based on Wightman's axioms.
*

R. Haag, *Local Quantum Physics: Fields, Particles, Algebras*, Springer 1996.

Manny: * The most explanatory treatese on QFT that I have found. It
has detailed discussions and describes the necessary background
nicely, but for detailed physics arguments and proofs (when
available) you need to follow the references. It addresses
foundational issues that are rarely clarified elsewhere.
*

- Perturbative Algebraic Quantum Field Theory:
K. Fredenhagen and K. Rejzner, *Perturbative Algebraic Quantum Field Theory*,
arXiv:1208.1428.

R. Brunetti, M. Dütsch, K. Fredenhagen,
*Perturbative algebraic quantum field theory and the renormalization groups*,
Advances in Theoretical and Mathematical Physics, Volume **13**, Number 5 (2009), 1541-1599.

K. Rejzner, *Perturbative Algebraic Quantum Field Theory*, Springer 2016.

Manny: *The most complete compilation of definitions and results
relevant to perturbative AQFT. Its subtitle is "An introduction
for mathematicians", but it is not an introduction, you have to be
comfortable in an exceptionally broad range of mathematics to make
sense of it, and you better know what physics motivates this. TMarch 21o
read and understand what's going on here you need ready access to
a massive amount of other literature for the details, including
proofs of many important results. The referenced proofs are too
often not mathematically satisfactory or complete without
backtracking through further literature. Annoyingly, there are
multiple references to unpublished results of the author. On the
other hand, the field has progressed a lot in the last decade, and
while it is far from settled, this is the most complete overview
of what has been accomplished. It is a commendable if frustrating
book. *

R. Brunetti, K. Fredenhagen, R. Verch, *The Generally Covariant Locality Principle -- A New Paradigm for
Local Quantum Field Theory*, Communications in Mathematical Physics **237**, 31-68 (2003)
arXiv:0112041.

Manny: *This describes the modern AQFT framework for field
theory on a general curved space-time background. The covariance
is implemented with a category-theoretical formalism. Pay close
attention to "relative Cauchy evolution", I found the underlying
idea to be universally useful.*

K. Fredenhagen and K. Rejzner, *Quantum field theory on curved spacetimes: Axiomatic framework and examples*,
Journal of Mathematical Physics **57** (2016).

- Mathematical Renormalization:
Borcherds, R., *Renormalization and quantum field theory,*
Algebra & Number Theory, Vol. **5**, Nr 5 (2011), 627-658.