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I don't have enough reputation to comment but please note that Weinberg uses the (-,+,+,+) metric, which means you need a change of sign in $$e^{iP.x}$$. The field transforms as, with $x'=x+a$, $$\phi'(x')=U^{-1}(a)\phi(x')U(a)=\phi(x)$$. For (+,-,-,-) signature this is implemented by $U(a)=e^{iP.x}$, see Peskin & Schroeder, page 26 for example. In ...

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Time-ordering is needed if the Hamiltonians $H(t^{\prime})$ and $H(t^{\prime\prime})$ at different times do not commute. Example: If the Hamiltonian is $$H(t) ~=~ \left\{\begin{array}{ccl} \color{Red}{\diamondsuit}&\text{if}& t<0, \cr \clubsuit &\text{if}& t>0, \end{array} \right.$$ then the non-time-ordered product $(\int\! dt ~H(t)... 3 You have to keep in mind that an integral is (more or less) a sum, and therefore in a double or multiple integral, you have cross terms that have to be correctly time-ordered. The usual Riemann integral does not take in account the non-commuting nature of its integrand, so you have to make it manifest with the time-ordering symbol$\mathcal{T}. Imagine for ... 3 Consider $$\int_0^t dt'\int_0^{t}dt'' \mathcal{T}\left(H(t')H(t'')\right)$$ \begin{align} &= \int_0^t dt'\int_0^{t'}dt'' \mathcal{T}\left(H(t')H(t'')\right) \\ & \qquad +\int_0^t dt'\int_{t'}^{t}dt'' \mathcal{T}\left(H(t')H(t'')\right) \\&= \int_0^t dt'\int_0^{t'}dt'' H(t')H(t'') \\ & \qquad +\int_0^t dt'\int_{t'}^{t}dt''H(t'')H(t') \end{... 2 The evolution operator generated by a time-dependent Hamiltonian has two parameters, not just one (i.e. the initial and final time). Let's denote such evolution by(U(t,s))_{(t,s)\in\mathbb{R}^2}$. Formally, we can write$U(t,s)=e^{-i\int_s^t H(\tau)d\tau}$(or with time-ordering if you want), but this has to be intended as follows.$(U(t,s))_{(t,s)\in\...

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Gauge symmetry is actually not spontaneously broken in the Higgs mechanism; this is a common misconception. See What role does "spontaneously symmetry breaking" played in the "Higgs Mechanism"?. Therefore the Mermin-Wagner theorem does not apply to the Higgs mechanism, and the Higgs mechanism is possible in 1+1D.

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A part of the supersymmetry algebra is $$\{Q_a,~{\bar Q}_{\dot b}\}~=~-2i\sigma^\mu_{a\dot b}\partial_\mu$$ which is a momentum operator $p_\mu~=~-i\partial_\mu$. The graded Lie algebra $g~=~h~+~k$ $$[h,~h]~\subset~h,~[h,~k]~\subset~k,~\{k,~k\}~\subset~h,$$ where the last of these contains the above anti-commutator. This model has chiral symmetry. It ...

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(Note that I am only starting to study these works and I may be wrong on some points.) The paper you are quoting is indeed providing a full definition of (type II and heterotic) superstring theory (type I is missing), valid at the quantum level and for both the NS and R sectors. The definition is basically following the construction of Zwiebach (arxiv:hep-...

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