# Why does $S$-matrix theory end up being a covariant formalism when it is not obvious that it is?

A principle of QFT that is frequently invoked, repeated, and potentially subject to rigorous verification is that the theory in question must exhibit Lorentz covariance and be invariant under the Poincaré group. Nevertheless, upon closer inspection of specific facts, it becomes less evident that particular formalisms are covariant explicitly:

1. In QFT, a Hamiltonian, which does not use time on an equal footing with the other space coordinates, is used. This contrasts with the ''Hamiltonianish'' formalism of De Donder-Weyl which resorts to a "Hamiltonianish" function where spatial and temporal coordinates are treated symmetrically.

2. Second, the $$S$$-matrix itself uses time-ordered products ($$\hat{T}[\ ]$$) and a Hamiltonian that treats time distinctly.

3. Feynman's diagrams are clearly associated with the Dyson series that uses both time-ordered products and Hamiltonian (which do not look like legitimately covariant objects).

Maybe I'm missing something, but I have yet to see a formal argument or an explicit demonstration that shows that, while not explicitly violating the covariance, the above three objections only appear to violate it. Given this situation, I attempted to find some valid reasons why the theory ultimately ends up being covariant. For example, when I consider the Penrose diagram of the Minkowski spacetime, it is clear that the past timelike infinity and the future timelike infinity are very special, and thus possibly the limit used to define the matrix $$\hat{S}$$:

$$\hat{S} = \lim_{t_2 \to +\infty} \lim_{t_1 \to -\infty} \langle \Psi_f |\hat{U}(t_2,t_1)|\Psi_0\rangle$$

with $$\hat{U}(t_2,t_1) = \hat{T}\left[ \exp\left( i\int_{t_1}^{t_2} V(t)\text{d}t \right)\right]$$ does not depend on the end of the coordinate system given how special the past and future timelike infinities are in Minkowski spacetime. Possibly this also applies to the Dyson series for the $$\hat{S}$$ array:

$$\hat{S} = \sum_{n=0}^\infty \frac{(-i)^n}{n!} \int_{-\infty}^{\infty} \hat{T}[\hat{H}(t_1)\dots\hat{H}(t_n)]\text{d}t_1\dots\text{d}t_n$$

Where the limits of integration end up being taken over the past and future timelike infinities.

• Have you read Weinberg? This is explained in the first 4-5 chapters (can't remember which exact chapter it is) Feb 26 at 22:56

The time ordered "$$T$$" product is formally Lorentz covariant because the fields commute at spacelike separation --- so "$$t$$" is in no way being singled out. I say "formally" because Schwinger terms involving space derivatives of $$\delta(x)$$ can violate the covariance. If so, one has to fix them up to get "$$T^*$$" orderings: See the book Current Algebra and Anomalies, by Sam Treiman, Roman Jackiw, Bruno Zumino, Edward Witten (Princeton University Press 1985). Path integrals usually give $$T^*$$ ordering and hence Lorentz covariant expressions in the S-matrix.