/sci/ - Science & Math

>>11352892
>correct
No. Stability is determined by the local minima of the energy functional [math]S:\mathcal{V}\rightarrow\mathbb{R}[/math], where [math]\mathcal{V} = \mathcal{I}\times \mathcal{E}[/math] is the space of configurations. If we write [math]S_E:\mathcal{I}\rightarrow \mathbb{R}[/math] for each [math]E\in\mathcal{E}[/math], decays [math]A\rightarrow B[/math] between critical points [math]A,B\in\mathcal{I}[/math] of [math]S_E[/math], at a [math]fixed~E\in\mathcal{E}[/math], can be understood as the meta/instability of [math]A[/math] with a unstable manifold of dim [math]> 0[/math] into [math]B[/math]. Over a certain range [math]E\in D\subset\mathcal{E}[/math] this may be the case, until a phase transition occurs at [math]E_0 \in \partial D[/math] where the unstable manifold from [math]A[/math] into [math]B[/math] becomes stable. This means that across this phase transition, the configuration [math]A[/math] is more stable than [math]B[/math]. If [math]S_\mathcal{E}[/math] is Cerf then these phase transitions are characterized by the Floer cohomology groups [math]\hat{H}^\ast_E(\mathcal{I}),\check{ H}^\ast_E(\mathcal{I}),\overline{H}^\ast_E(\mathcal{I})[/math] as [math]E\in\mathcal{E}[/math] varies.
Strange matter is entirely reasonable, as the formation of stars (exotic or otherwise) constitute such a phase transition characterized by these cohomology groups, hence there is no reason to expect top/down quarks to still be the lowest energy. It is on you to prove that, in the case of SM where [math]\mathcal{I}[/math] is an infinite dimensional [math]SU(3)\times SU(2)\times U(1)[/math]-equivariant jet bundle, that these Floer groups are insensitive to these star-formation phase transitions.

>>11316088
>not comfy
It's comfy when there aren't any schizos/cranks shitting it up.

>>11266572
There is no "time operator" in QM. In non-relativistic theories "time" is merely the parameter in a physical trajectory on the symplectic manifold [math](M,\omega)[/math]. If [math]M \cong T^*Q[/math] for some configuration space [math]Q[/math] then [math]M[/math] is locally described by [math](p,q)[/math], namely the position and momentum for which [math]\omega = \sum dq \wedge dp[/math]. Prequantization promotes [math]C^\infty(M)[/math] to operators onto Hilbert sections [math]\Gamma(M,B)[/math] of the prequantum bundle [math]B\rightarrow M[/math], wherein time never explicitly appears.
I know there are elementary intro texts that "treat" [math]t[/math] like an operator to obtain an uncertainty principle for time and the energy but it's merely a property of the time evolution operator [math]U(t) = \exp (-i Ht)[/math]. Promoting [math]t[/math] to multiplication by [math]t[/math] on lifts of physical trajectories into [math]\Gamma(M,B)[/math] is inherently [math]not[/math] part of quantization.
>>11266593
Again, in relativistic QM [math]M \cong T^*Q[/math] for a Minkowski manifold [math]Q[/math], which is described locally as a tuple of 4-vectors [math](p,q)[/math]. Quantization occurs for (proper) time in this case, yes, but there is still no reason to combine the two. Essentially, [math]q = (q_0=t,{\bf q})[/math] promotes to generators of the Lie algebra of translations by the 4-momentum [math](p_0,{\bf p})[/math], which is a linear space where any multiple of [math]q,q_0[/math] can be expressed in terms of linear combinations of [math]q_0,q[/math].
>it wouldn't have a real-valued eigenvalue
Since translations are unitary, [math]q[/math] is Hermitian. Any combination (linear or otherwise) is then also Hermitian so their eigenvalues are also real. Please finish reading Griffith first, sweetie.

>>11154170
In general slashed quantities appear in observable scattering amplitudes like [math]\operatorname{tr}\not p[/math]. Note that the gamma matrices furnishes a spinor representation of [math]\mathfrak{sp}(1,3)= \operatorname{Isom}\mathbb{M}[/math], this means that the map [math]\operatorname{Tr}^\text{Spin} = \operatorname{tr}\gamma \cdot[/math] is a map that, intuitively, averages over the spin degree of freedom. You can think of [math]\operatorname{Tr}^\text{Spin}[/math] as a "supertrace" of sorts.
>>11154271
>Or you just mean that the kernel is isomorphic to A?
Yep. In fact, projective modules are exactly those that enter as a semidirect product factor in a module [math]B[/math], at least locally in terms of its generators. As a direct analogy, fibre bundles fit into the exact sequence [math]0\rightarrow F\rightarrow E\xrightarrow{\pi} B\rightarrow 0[/math] with [math]F[/math] the fibre space, and vector bundles are always locally trivializable with [math]\pi^{-1}U \cong F \times B[/math] for [math]U \subset B[/math].