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Register a new userDoes the first part of the second law also imply its second part?
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Authors
N. D. Hari Dass
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Sommerfeld called the first part of the second law to be the entropy axiom, which is about the existence of the state function entropy. It was usually thought that the second part of the second law, which is about the non-decreasing nature of entropy of thermally isolated systems, did not follow from the first part. In this note, we point out the surprise that the first part in fact implies the second part.
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Within the geometrical framework developed in arXiv:0705.2362, the problem of minimality for constrained calculus of variations is analysed among the class of differentiable curves. A fully covariant representation of the second variation of the action functional, based on a suitable gauge transformation of the Lagrangian, is explicitly worked out. Both necessary and sufficient conditions for minimality are proved, and are then reinterpreted in terms of Jacobi fields.
We provide a novel perspective on regularity as a property of representations of the Weyl algebra. In Part I, we critiqued a proposal by Halvorson [2004, Complementarity of representations in quantum mechanics, Studies in History and Philosophy of Modern Physics 35(1), pp. 45--56], who advocates for the use of the non-regular position and momentum representations of the Weyl algebra. Halvorson argues that the existence of these non-regular representations demonstrates that a quantum mechanical particle can have definite values for position or momentum, contrary to a widespread view. In this sequel, we propose a justification for focusing on regular representations, pace Halvorson, by drawing on algebraic methods.
We provide a novel perspective on regularity as a property of representations of the Weyl algebra. We first critique a proposal by Halvorson [2004, Complementarity of representations in quantum mechanics, Studies in History and Philosophy of Modern Physics 35(1), pp. 45--56], who argues that the non-regular position and momentum representations of the Weyl algebra demonstrate that a quantum mechanical particle can have definite values for position or momentum, contrary to a widespread view. We show that there are obstacles to such an intepretation of non-regular representations. In Part II, we propose a justification for focusing on regular representations, pace Halvorson, by drawing on algebraic methods.
We consider the manipulation of multipartite entangled states in the limit of many copies under quantum operations that asymptotically cannot generate entanglement. As announced in [Brandao and Plenio, Nature Physics 4, 8 (2008)], and in stark contrast to the manipulation of entanglement under local operations and classical communication, the entanglement shared by two or more parties can be reversibly interconverted in this setting. The unique entanglement measure is identified as the regularized relative entropy of entanglement, which is shown to be equal to a regularized and smoothed version of the logarithmic robustness of entanglement. Here we give a rigorous proof of this result, which is fundamentally based on a certain recent extension of quantum Steins Lemma proved in [Brandao and Plenio, Commun. Math. 295, 791 (2010)], giving the best measurement strategy for discriminating several copies of an entangled state from an arbitrary sequence of non-entangled states, with an optimal distinguishability rate equal to the regularized relative entropy of entanglement. We moreover analyse the connection of our approach to axiomatic formulations of the second law of thermodynamics.
We present in Part II the description of the internal degrees of freedom of fermions by the superposition of odd products of the Clifford algebra elements, either $gamma^a$s or $tilde{gamma}^a$s, which determine with their oddness the anticommuting properties of the creation and annihilation operators of the second quantized fermion fields in even $d$-dimensional space-time, as we do in Part I of this paper by the Grassmann algebra elements $theta^a$s and $frac{partial}{partial theta_a}$s. We discuss: {bf i.} The properties of the two kinds of the odd Clifford algebras, forming two independent spaces, both expressible with the Grassmann algebra of $theta^{a}$s and $frac{partial}{partial theta_{a}}$s. {bf ii.} The freezing out procedure of one of the two kinds of the odd Clifford objects, enabling that the remaining Clifford objects determine with their oddness in the tensor products of the finite number of the Clifford basis vectors and the infinite number of momentum basis, the creation and annihilation operators carrying the family quantum numbers and fulfilling the anticommutation relations of the second quantized fermions: on the vacuum state, and on the whole Hilbert space defined by the sum of infinite number of Slater determinants of empty and occupied single fermion states. {bf iii.} The relation between the second quantized fermions as postulated by Dirac and the ones following from our Clifford algebra creation and annihilation operators, what offers the explanation for the Dirac postulates.
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