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== Attitudes and interpretation ==

== Attitudes and interpretation ==

The early formulators of QED and other quantum field theories were, as a rule, dissatisfied with this state of affairs. It seemed illegitimate to do something tantamount to subtracting infinities from infinities to get finite answers.

The early formulators of QED and other quantum field theories were, as a rule, dissatisfied with this state of affairs. It seemed illegitimate to do something tantamount to subtracting infinities from infinities to get finite answers.

The early formulators of QED and other quantum field theories were, as a rule, dissatisfied with this state of affairs. It seemed illegitimate to do something tantamount to subtracting infinities from infinities to get finite answers.

The early formulators of QED and other quantum field theories were, as a rule, dissatisfied with this state of affairs. It seemed illegitimate to do something tantamount to subtracting infinities from infinities to get finite answers.

Freeman Dyson argued that these infinities are of a basic nature and cannot be eliminated by any formal mathematical procedures, such as the renormalization method.

Freeman Dyson argued that these infinities are of a basic nature and cannot be eliminated by any formal mathematical procedures, such as the renormalization method.

Dirac's criticism was the most persistent. As late as 1975, he was saying:

Dirac's criticism was the most persistent. As late as 1975, he was saying:

狄拉克的批评是最持久的。直到1975年，他还在说:

狄拉克的批评是最持久的。直到1975年，他仍表示:

  Most physicists are very satisfied with the situation. They say: 'Quantum electrodynamics is a good theory and we do not have to worry about it any more.' I must say that I am very dissatisfied with the situation because this so-called 'good theory' does involve neglecting infinities which appear in its equations, ignoring them in an arbitrary way. This is just not sensible mathematics. Sensible mathematics involves disregarding a quantity when it is small – not neglecting it just because it is infinitely great and you do not want it!

  Most physicists are very satisfied with the situation. They say: 'Quantum electrodynamics is a good theory and we do not have to worry about it any more.' I must say that I am very dissatisfied with the situation because this so-called 'good theory' does involve neglecting infinities which appear in its equations, ignoring them in an arbitrary way. This is just not sensible mathematics. Sensible mathematics involves disregarding a quantity when it is small – not neglecting it just because it is infinitely great and you do not want it!

大多数物理学家对这种情况非常满意。他们说: 量子电动力学理论是一个很好的理论，我们再也不用担心它了我必须说，我对这种情况非常不满意，因为这种所谓的“好理论”确实涉及到忽视方程式中出现的无穷大，以一种武断的方式忽视它们。这根本不是合理的数学。明智的数学涉及到忽略一个小量——不要仅仅因为它无限大而你不想要它就忽略它！

大多数物理学家对这种情况非常满意。他们说: ‘量子电动力学理论是一个很好的理论，我们再也不用担心它了’。但我必须说，我对这种情况非常不满意，因为这种所谓的“好理论”确实有涉及忽视方程式中出现的无穷大，且以一种武断的方式忽视它们。这根本不是合理的数学。合理的数学涉及对微小量的忽略，而不是仅仅因为一个量无限大，你不想要它就可以忽略它！

  The shell game that we play is technically called 'renormalization'. But no matter how clever the word, it is still what I would call a dippy process! Having to resort to such hocus-pocus has prevented us from proving that the theory of quantum electrodynamics is mathematically self-consistent. It's surprising that the theory still hasn't been proved self-consistent one way or the other by now; I suspect that renormalization is not mathematically legitimate.

  The shell game that we play is technically called 'renormalization'. But no matter how clever the word, it is still what I would call a dippy process! Having to resort to such hocus-pocus has prevented us from proving that the theory of quantum electrodynamics is mathematically self-consistent. It's surprising that the theory still hasn't been proved self-consistent one way or the other by now; I suspect that renormalization is not mathematically legitimate.

我们玩的这个骗局在技术上叫做“重整化”。但是不管这个词多么聪明，它仍然是我所说的一个迷糊的过程！不得不求助于这样的骗术使我们无法证明量子电动力学理论在数学上是自洽的。令人惊讶的是，到目前为止，这个理论仍然没有以这样或那样的方式被证明是自洽的; 我怀疑重整化在数学上是不合理的。

我们玩的这个骗局在技术上叫做“重整化”。但是不管这个词多么聪明，它仍然是我所说的一个含糊的过程！不得不求助于这样的骗术阻碍了我们证明量子电动力学理论在数学上是自洽的脚步。令人惊讶的是，到目前为止，这个理论仍然没有以这样或那样的方式被证明是自洽的; 我怀疑重整化在数学上是不正当的。

Feynman was concerned that all field theories known in the 1960s had the property that the interactions become infinitely strong at short enough distance scales. This property called a Landau pole, made it plausible that quantum field theories were all inconsistent. In 1974, Gross, Politzer and Wilczek showed that another quantum field theory, quantum chromodynamics, does not have a Landau pole. Feynman, along with most others, accepted that QCD was a fully consistent theory.

Feynman was concerned that all field theories known in the 1960s had the property that the interactions become infinitely strong at short enough distance scales. This property called a Landau pole, made it plausible that quantum field theories were all inconsistent. In 1974, Gross, Politzer and Wilczek showed that another quantum field theory, quantum chromodynamics, does not have a Landau pole. Feynman, along with most others, accepted that QCD was a fully consistent theory.

费曼担心，在20世纪60年代所有已知的场理论都有这样的特性: 在足够短的距离尺度上，相互作用会变得无限强烈。这个性质被称为兰道极，使得量子场理论全部不一致的说法似乎是可信的。1974年，Gross，Politzer 和 Wilczek 证明了另一个量子场论，量子色动力学，并没有兰道极点。费曼和其他大多数人一样，承认 QCD 是一个完全一致的理论。

费曼担心，在20世纪60年代所有已知的场理论都有这样的特性: 在足够短的距离尺度上，相互作用会变得无限强。这个性质被称为Landau pole，它使量子场理论完全不一致的说法变得可信。1974年，Gross，Politzer 和 Wilczek 证明了另一个量子场论（即量子色动力学），并没有Landau pole。费曼和其他大多数人一样，承认量子色动力学是一个完全一致的理论。

The general unease was almost universal in texts up to the 1970s and 1980s. Beginning in the 1970s, however, inspired by work on the renormalization group and effective field theory, and despite the fact that Dirac and various others—all of whom belonged to the older generation—never withdrew their criticisms, attitudes began to change, especially among younger theorists. Kenneth G. Wilson and others demonstrated that the renormalization group is useful in statistical field theory applied to condensed matter physics, where it provides important insights into the behavior of phase transitions. In condensed matter physics, a physical short-distance regulator exists: matter ceases to be continuous on the scale of atoms. Short-distance divergences in condensed matter physics do not present a philosophical problem since the field theory is only an effective, smoothed-out representation of the behavior of matter anyway; there are no infinities since the cutoff is always finite, and it makes perfect sense that the bare quantities are cutoff-dependent.

The general unease was almost universal in texts up to the 1970s and 1980s. Beginning in the 1970s, however, inspired by work on the renormalization group and effective field theory, and despite the fact that Dirac and various others—all of whom belonged to the older generation—never withdrew their criticisms, attitudes began to change, especially among younger theorists. Kenneth G. Wilson and others demonstrated that the renormalization group is useful in statistical field theory applied to condensed matter physics, where it provides important insights into the behavior of phase transitions. In condensed matter physics, a physical short-distance regulator exists: matter ceases to be continuous on the scale of atoms. Short-distance divergences in condensed matter physics do not present a philosophical problem since the field theory is only an effective, smoothed-out representation of the behavior of matter anyway; there are no infinities since the cutoff is always finite, and it makes perfect sense that the bare quantities are cutoff-dependent.

直到20世纪70年代和80年代，这种普遍的不安在文本中几乎是普遍存在的。然而，从20世纪70年代开始，受到重整化群理论和有效场理论工作的启发，尽管狄拉克和其他各种人---- 他们都属于老一辈---- 从未收回他们的批评，但态度开始改变，尤其是在年轻的理论家中。肯尼斯·威尔森和其他人证明了重整化群在统计领域理论应用于凝聚态物理学中是有用的，它提供了对相变行为的重要见解。在21凝聚态物理学，存在一个物理的短距离调节器: 物质不再是连续的原子规模。凝聚态物理学中的短距离分歧并不构成哲学问题，因为场论只是对物质行为的一种有效的、平滑的表示; 因为截止值总是有限的，所以没有无限性，而且光裸量是截止值依赖的，这是完全合理的。

直到20世纪70年代和80年代，这种普遍的不安在文献中几乎是普遍存在的。然而，从20世纪70年代开始，尽管狄拉克和其他各种人---- 那些老一辈的理论家---- 从未收回他们的批评，受到重整化群理论和有效场理论工作的启发，人们的态度开始改变，尤其是在年轻的理论家中。肯尼斯·威尔森和其他人证明了重整化群对凝聚态物理学中统计场理论的应用有帮助，因为它对相变行为提供了重要的见解。在凝聚态物理学，存在一个物理的短距离调节器:在原子规模下，物质不再是连续的。凝聚态物理学中的短距离发散并不构成哲学问题，因为场论实际也只是对物质行为的一种有效的、平滑的理想表示; 因为截止值总是有限的，所以没有无限性，而且裸量依赖于截止值也是完全合理的。

 

If QFT holds all the way down past the Planck length (where it might yield to string theory, causal set theory or something different), then there may be no real problem with short-distance divergences in particle physics either; all field theories could simply be effective field theories. In a sense, this approach echoes the older attitude that the divergences in QFT speak of human ignorance about the workings of nature, but also acknowledges that this ignorance can be quantified and that the resulting effective theories remain useful.

If QFT holds all the way down past the Planck length (where it might yield to string theory, causal set theory or something different), then there may be no real problem with short-distance divergences in particle physics either; all field theories could simply be effective field theories. In a sense, this approach echoes the older attitude that the divergences in QFT speak of human ignorance about the workings of nature, but also acknowledges that this ignorance can be quantified and that the resulting effective theories remain useful.

如果 QFT 能一直保持到普朗克长度以下(在那里它可能会产生弦论、因果集合论或其他不同的理论) ，那么粒子物理学中的短距离分歧可能也不存在真正的问题; 所有场论都可能只是有效场论。在某种意义上，这种方法呼应了以前的观点，即量子力学中的分歧说明了人类对自然规律的无知，但也承认这种无知是可以量化的，由此产生的有效理论仍然是有用的。

如果量子场论能一直保持成立到普朗克长度以下(在那里它可能会产生弦论、因果集合论或其他不同的理论) ，那么粒子物理学中的短距离发散可能也不存在实质的问题; 所有场论都可能是有效场论。在某种意义上，这种方法呼应了以前的态度，即量子力学中的发散说明了人类对自然运作规律的无知，但也承认这种无知是可以量化的，且由此产生的有效理论仍然是有用的。

Be that as it may, Salam's remark in 1972 seems still relevant

Be that as it may, Salam's remark in 1972 seems still relevant

  Field-theoretic infinities — first encountered in Lorentz's computation of electron self-mass — have persisted in classical electrodynamics for seventy and in quantum electrodynamics for some thirty-five years. These long years of frustration have left in the subject a curious affection for the infinities and a passionate belief that they are an inevitable part of nature; so much so that even the suggestion of a hope that they may, after all, be circumvented — and finite values for the renormalization constants computed — is considered irrational. Compare Russell's postscript to the third volume of his autobiography  The Final Years, 1944–1969 (George Allen and Unwin, Ltd., London 1969), p. 221:

  Field-theoretic infinities — first encountered in Lorentz's computation of electron self-mass — have persisted in classical electrodynamics for seventy and in quantum electrodynamics for some thirty-five years. These long years of frustration have left in the subject a curious affection for the infinities and a passionate belief that they are an inevitable part of nature; so much so that even the suggestion of a hope that they may, after all, be circumvented — and finite values for the renormalization constants computed — is considered irrational. Compare Russell's postscript to the third volume of his autobiography  The Final Years, 1944–1969 (George Allen and Unwin, Ltd., London 1969), p. 221:

场论的无穷大首次出现在洛伦兹计算电子自质量的过程中，它在经典电动力学中已经存在了七十年，在量子电动力学中已经存在了35年。这么多年的挫折使得这个课题对无穷大产生了一种奇怪的感情，并且热切地相信它们是自然界不可避免的一部分; 以至于即使有希望它们毕竟可以被绕过ーー计算出的重整化常数的有限值ーー都被认为是不合理的。将罗素的附言与他的自传《最后的岁月，1944-1969》(乔治 · 艾伦和安文出版社，伦敦，1969年)的第三卷相比较，第221页:

场论的无穷大首次出现在洛伦兹对电子自质量的计算过程中，它在经典电动力学中已经存在了七十年，在量子电动力学中也已存在了35年。这么多年的挫折使得研究这个课题的人对无穷大产生了一种奇怪的感情，并且热切地相信它们是自然界不可避免的一部分; 以至于即使他们有可能避开了正解，有限的重整化常数可被计算--这样的希望都被认为是不合理的。将罗素的附言与他的自传《最后的岁月，1944-1969》(乔治 · 艾伦和安文出版社，伦敦，1969年)的第三卷相比较，于第221页:

  In the modern world, if communities are unhappy, it is often because they have ignorances, habits, beliefs, and passions, which are dearer to them than happiness or even life. I find many men in our dangerous age who seem to be in love with misery and death, and who grow angry when hopes are suggested to them. They think hope is irrational and that, in sitting down to lazy despair, they are merely facing facts.

  In the modern world, if communities are unhappy, it is often because they have ignorances, habits, beliefs, and passions, which are dearer to them than happiness or even life. I find many men in our dangerous age who seem to be in love with misery and death, and who grow angry when hopes are suggested to them. They think hope is irrational and that, in sitting down to lazy despair, they are merely facing facts.

In QFT, the value of a physical constant, in general, depends on the scale that one chooses as the renormalization point, and it becomes very interesting to examine the renormalization group running of physical constants under changes in the energy scale.  The coupling constants in the Standard Model of particle physics vary in different ways with increasing energy scale: the coupling of quantum chromodynamics and the weak isospin coupling of the electroweak force tend to decrease, and the weak hypercharge coupling of the electroweak force tends to increase.  At the colossal energy scale of 10¹⁵ GeV (far beyond the reach of our current particle accelerators), they all become approximately the same size (Grotz and Klapdor 1990, p. 254), a major motivation for speculations about grand unified theory.  Instead of being only a worrisome problem, renormalization has become an important theoretical tool for studying the behavior of field theories in different regimes.

In QFT, the value of a physical constant, in general, depends on the scale that one chooses as the renormalization point, and it becomes very interesting to examine the renormalization group running of physical constants under changes in the energy scale.  The coupling constants in the Standard Model of particle physics vary in different ways with increasing energy scale: the coupling of quantum chromodynamics and the weak isospin coupling of the electroweak force tend to decrease, and the weak hypercharge coupling of the electroweak force tends to increase.  At the colossal energy scale of 10¹⁵ GeV (far beyond the reach of our current particle accelerators), they all become approximately the same size (Grotz and Klapdor 1990, p. 254), a major motivation for speculations about grand unified theory.  Instead of being only a worrisome problem, renormalization has become an important theoretical tool for studying the behavior of field theories in different regimes.

在 QFT 中，一个物理常数的值，一般来说，取决于我们选择的重整化点的尺度，在能量尺度变化的情况下，研究重整化群物理常数的运行变得非常有趣。粒子物理标准模型中的耦合常数随着能量的增加而以不同的方式变化: 量子色动力学的耦合和电弱力的弱同位旋耦合趋于减小，电弱力的弱超荷耦合趋于增加。在10 < sup > 15 GeV 的巨大能量范围内(远远超出我们现有的粒子加速器的能量范围) ，它们都变得大致相同(Grotz 和 Klapdor 1990，p. 254) ，这是推测大统一理论的主要动机。重整化已经不再是一个令人担忧的问题，而是成为研究不同区域中场理论行为的一个重要理论工具。

在量子场论中，一个物理常数的值，一般来说，取决于我们选择的重整化点的尺度，在能量尺度变化的情况下，研究重整化群物理常数的运行变得非常有趣。粒子物理标准模型中的耦合常数随着能量的增加而以不同的方式变化: 量子色动力学的耦合和电弱力的弱同位旋耦合趋于减小，电弱力的弱超荷耦合趋于增加。在10¹⁵ GeV 的巨大能量尺度内(远远超出我们现有的粒子加速器的能量范围) ，它们都变得大致相同(Grotz 和 Klapdor 1990，p. 254) ，这是推测大统一理论的主要动机。重整化已经不再是一个令人担忧的问题，而是成为研究不同区域中场理论行为的一个重要理论工具。

If a theory featuring renormalization (e.g. QED) can only be sensibly interpreted as an effective field theory, i.e. as an approximation reflecting human ignorance about the workings of nature, then the problem remains of discovering a more accurate theory that does not have these renormalization problems.  As Lewis Ryder has put it, "In the Quantum Theory, these [classical] divergences do not disappear; on the contrary, they appear to get worse.  And despite the comparative success of renormalisation theory, the feeling remains that there ought to be a more satisfactory way of doing things."

If a theory featuring renormalization (e.g. QED) can only be sensibly interpreted as an effective field theory, i.e. as an approximation reflecting human ignorance about the workings of nature, then the problem remains of discovering a more accurate theory that does not have these renormalization problems.  As Lewis Ryder has put it, "In the Quantum Theory, these [classical] divergences do not disappear; on the contrary, they appear to get worse.  And despite the comparative success of renormalisation theory, the feeling remains that there ought to be a more satisfactory way of doing things."

如果一个具有重整化特征的理论(例如:。量子电动力学(QED)只能理性地解释为一种有效场理论，即。作为反映人类对自然规律的无知的近似值，那么发现一个更精确的理论，而不存在这些重整化问题的问题仍然存在。正如路易斯 · 莱德所说，“在量子理论中，这些[经典]分歧并没有消失; 相反，它们似乎变得更糟。尽管重整化理论相对成功，但人们仍然认为，应该有一种更令人满意的做事方式。”

如果一个具有重整化特征的理论(例如:量子电动力学(QED))只能合理地被解释为一种有效场理论，即作为反映人类对自然规律的无知的近似反应，那么问题仍在于发现一个更精确的，不存在这些重整化问题的理论。正如路易斯 · 莱德所说，“在量子理论中，这些[经典]发散并没有消失; 相反，它们似乎变得更糟。尽管重整化理论相对成功，但人们仍然认为，应该有一种更令人满意的处理方法。”