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Retrieving "Field" from the archives

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  1. Designated Hitter

    Linked via "field"

    The Designated Hitter (DH) rule is a specific regulation in the sport of baseball ($sport$), primarily used in certain professional leagues, that permits a team to substitute a player to bat in place of the pitcher without removing the pitcher from the defensive lineup. This substitution is purely offensive, meaning the Designated Hitter does not play a defensive position on the field. The rule's primary effect is to increase [offensive production](/entries/offensive-productio…

  2. Jordan Algebras

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    Definition and Axiomatics
    A Jordan algebra $J$ over a field $F$ (often $\mathbb{R}$ or $\mathbb{C}$, though investigations over fields with characteristic 3 have shown surprising resilience) is a vector space equipped with a binary product, denoted by $\circ$, that satisfies two key axioms:
    Commutativity: For all $x, y \in J$:

  3. Lagrangian Density

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    The Lagrangian density ($\mathcal{L}$) is a scalar function of the generalized coordinates (field) and their time and spatial derivatives, central to the formulation of classical field theory and quantum field theory. It serves as the foundation for deriving the equations of motion for a physical system through the principle of least action. Unlike the simpler Lagrangian used in [particle mechanics](/entrie…

  4. Lagrangian Density

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    (Note: Indices are contracted over all independent fields $\phi_a$).
    While this canonical tensor is mathematically sound, it often lacks the desired physical properties, such as symmetry ($T^{\mu\nu} = T^{\nu\mu}$) or tracelessness ($T^\mu\mu = 0$), which are necessary for modern physical interpretations (e.g., energy density being positive). The true physical tensor, often referred to as the Believed Stress-Energy Tensor, $T^{\mu\nu}{\text{physical}}$, is obtained by adding specific four-divergences to the canonical expression until the desired symmetries are m…

  5. Spontaneous Symmetry Breaking

    Linked via "field ($\phi$)"

    Theoretical Formalism
    The concept of SSB is mathematically formalized through the analysis of the Lagrangian density ($\mathcal{L}$)} of the system, which remains invariant under a specific set of symmetry transformations (e.g., rotations, phase shifts, or local gauge transformations). However, the vacuum expectation value ($\langle \phi \rangle$)} of the relevant field ($\phi$)} does not transform to zero under these operations, leading to the broken symmetry…