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Comments to the question (v3): I) The Gurtin-Tonti bi-local method [which OP mentions in an example; see also Section II below] of pairing opposite times $t\leftrightarrow (t_f-t_i)-t$ (hidden inside a convolution) is an artificial trick from a fundamental physics point of view, unless further justified. Why would such correlations into the past/future take ...


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Momentum is be conserved iff the Hamiltonian has translational symmetry. Usual boundary conditions such as homogeneous Dirichlet or Neumann conditions don't allow for such symmetry. But there still are specific conditions, which do allow the Hamiltonian to have translational symmetry on the bounded domain: Born—von Karman boundary conditions. Thus in the ...


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The linear transformation is the following composition of linear maps: Go from $R^n$ to $T_m M$ using the natural identification Go from $T_m M$ to $T^*_m M$ with the symplectic form Go from $T^*_m M$ to $T_m M$ using the inverse isomorphism given by the metric Go again back to $R^n$ (here $M = R^n$ and $m = (q,p)$ By the way, this transformation of ...


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This would mean that the eigenbasis of a physical observable is not orthogonal. Is there an error in my derivation, and if not, how can this be understood physically? The set of eigenfunctions of $\hat p$ in the sense $$ \hat{p}\phi = p\phi $$ is sure to be orthogonal if they belong to a subset of $L^2((0,1))$ on which the operator $\hat{p}$ is ...


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Same as with the symplectic form: $\omega(v) = (u_\omega,v)$ defines the isomorphism between 1-forms and vector fields. When the metric is Euclidean the dual basis to an orthonormal basis corresponds to the basis itself.


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Frank Wilczek will edit The Princeton Companion to Physics, but unfortunately the anticipated publication date is in 2018.


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Suppose $g$ is the generator of a certain symmetry (i.e. the generating function of an infinitesimal canonical transformation) and you are interested to know how the observable $f$ changes after the "action" of $g$. In the Hamiltonian formalism the change is found to be $$\delta f \approx \epsilon\{f,g\}$$ which can be related to the time evolution of an ...


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I) Before we get to quantization and path integrals there are problems already at the classical level. The Legendre transformation is not well-defined without knowledge of the CCR. For instance if the CCRs for the complex bosonic scalar $\hat{\phi}$ and $\hat{\phi}^{\dagger}$ is zero, this would mean that OP's Hamiltonian density ${\cal H}$ is a pure ...


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The answer by Enucatl is satisfying enough. However, in the example $$P=q \cot(p),$$ $$Q=\ln \left (\frac{\sin(p)}{q}\right),$$ given in the question, it seems there is dimensional mismatching. The argument inside $\cot$ must be some $[p/(p_o)]$ where $p_o$ has dimensions of momentum and the argument of the logarithm must be $$q_o \frac{\sin(p/p'_o)}{q},$$ ...


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Comments to the question (v2): A field $\phi^{\alpha}:[t_i,t_f]\times \mathbb{R}^3\to \mathbb{R}$ is the field-theoretic version of a (generalized) position variable $q^i:[t_i,t_f]\to \mathbb{R}$ in point mechanics. Note that the physical position space $\mathbb{R}^3$ typically plays very different roles in field theory and in point mechanics.$^1$ ...


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The approximate equality is intended to remind you that Nakahara considers transformations with an infinitesimal parameter $\epsilon$ here, but you could as well take it as a full equality, it doesn't matter. The phase space is a cotangent bundle, where the coordinate $x$ are coordinates of the underlying manifold and the momenta $p$ lie in the cotangent ...



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