Relativistic probability densities for location

IF 1.1 Q3 PHYSICS, MULTIDISCIPLINARY
Joshua G. Fenwick, R. Dick
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引用次数: 0

Abstract

Imposing the Born rule as a fundamental principle of quantum mechanics would require the existence of normalizable wave functions ψ( x , t) also for relativistic particles. Indeed, the Fourier transforms of normalized k -space amplitudes ψ(k,t)=ψ(k)exp(−iωkt) yield normalized functions ψ( x , t) which reproduce the standard k -space expectation values for energy and momentum from local momentum (pseudo-)densities ℘ μ ( x , t) = (ℏ/2i)[ψ +( x , t)∂ μ ψ( x , t) − ∂ μ ψ +( x , t) · ψ( x , t)]. However, in the case of bosonic fields, the wave packets ψ( x , t) are nonlocally related to the corresponding relativistic quantum fields ϕ( x , t), and therefore the canonical local energy-momentum densities (x,t)=c0(x,t) and (x,t) differ from ℘ μ ( x , t) and appear nonlocal in terms of the wave packets ψ( x , t). We examine the relation between the canonical energy density (x,t) , the canonical charge density ϱ( x , t), the energy pseudo-density ˜(x,t)=c℘0(x,t) , and the Born density ∣ψ( x , t)∣2 for the massless free Klein–Gordon field. We find that those four proxies for particle location are tantalizingly close even in this extremely relativistic case: in spite of their nonlocal mathematical relations, they are mutually local in the sense that their maxima do not deviate beyond a common position uncertainty Δx. Indeed, they are practically indistinguishable in cases where we would expect a normalized quantum state to produce particle-like position signals, viz. if we are observing quanta with momenta p ≫ Δp ≥ ℏ/2Δx. We also translate our results to massless Dirac fields. Our results confirm and illustrate that the normalized energy density (x,t)/E provides a suitable measure for positions of bosons, whereas normalized charge density ϱ( x , t)/q provides a suitable measure for fermions.
位置的相对论概率密度
将玻恩规则作为量子力学的基本原理,需要相对论粒子也存在可归一化波函数ψ(x,t)。事实上,归一化k空间振幅ψ(k,t)=ψ(k)exp(−iωkt)的傅立叶变换产生归一化函数ψ(x,t),该函数再现了来自局部动量(伪)密度的能量和动量的标准k空间期望值℘ μ(x,t)=(ℏ/2i)[ψ+(x,t。然而,在玻色子场的情况下,波包ψ(x,t(x,t)=c0(x,t)和(x,t)不同于℘ μ(x,t),并且根据波包ψ(x,t)表现为非局部的。我们检验了正则能量密度之间的关系(x,t),规范电荷密度ϱ(x,t),能量伪密度~(x,t)=c℘0(x,t),以及无质量自由Klein-Gordon场的Born密度Şψ。我们发现,即使在这种极端相对论的情况下,粒子位置的这四个指标也非常接近:尽管它们之间存在非局部数学关系,但它们是相互局部的,因为它们的最大值不会偏离共同的位置不确定性Δx。事实上,在我们期望归一化量子态产生类似粒子的位置信号的情况下,它们实际上是不可区分的,即,如果我们观察到动量ṕΔp≥ℏ/2Δx。我们还将结果转化为无质量狄拉克场。我们的结果证实并说明了归一化能量密度(x,t)/E为玻色子的位置提供了一个合适的度量,而归一化电荷密度ϱ(x,t)/q为费米子提供了一种合适的度量。
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来源期刊
Journal of Physics Communications
Journal of Physics Communications PHYSICS, MULTIDISCIPLINARY-
CiteScore
2.60
自引率
0.00%
发文量
114
审稿时长
10 weeks
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