内容来源：https://www.quantamagazine.org/old-ghost-theory-of-quantum-gravity-makes-a-comeback-20251117/

内容总结：

【理论物理新突破】“幽灵粒子”理论重获生机，或破解引力量子化世纪难题

长期以来，引力作为人类最熟悉的基本力，其微观机制始终是物理学界的未解之谜。近日，一项曾被尘封的“二次引力”理论正在国际理论物理学界掀起复兴浪潮——该理论中曾令人闻之色变的“幽灵粒子”，可能正是连通引力与量子场论的关键钥匙。

传统粒子物理学遭遇的困境在于：当学者尝试用描述光子的量子场论方法解释引力时，计算总会出现无法消除的无穷大量。上世纪70年代，物理学家凯洛格·斯泰勒提出突破性解决方案：通过在爱因斯坦引力方程中引入两个与曲率平方相关的新项，成功实现了理论“重整化”，使引力首次在数学上获得与电磁力同等的微观描述资格。

然而这一理论代价巨大：方程中必然出现具有负能量的“幽灵粒子”。按照经典认知，这类粒子会引发真空崩溃和负概率悖论，导致理论被贴上“病态”标签而遭冷落四十年。

转机出现在最近十年。随着超对称粒子在大型强子对撞机实验中迟迟未现，以及弦理论突破受阻，学者们重新审视这一理论。荷兰拉德堡德大学布奥因凡特等学者发现：

1. 幽灵粒子实际寿命极短，尚未引发灾难便已湮灭
2. 通过调整概率计算规则，可确保所有物理过程保持因果律
3. 该理论预言的原初引力波强度与当前观测数据相容

更令人振奋的是，该理论中的无害标量粒子恰好符合宇宙暴胀模型需求。苏联物理学家斯塔罗宾斯基早在1980年就基于此提出首个暴胀模型，而最新研究表明其预言的时空涟漪可能被下一代望远镜捕获。

尽管学界仍对修改因果律等基础法则存有顾虑，但该理论近年引用量从年均20次猛增至150次，显示其正获得日益广泛的关注。正如马萨诸塞大学教授约翰·多诺霍所言：“这或许不是终极理论，但很可能揭示了现实的一个自洽层面——在某个尺度以下，无论时空本质是弦、圈还是其他结构，我们都能用这套框架做出准确预测。”

斯泰勒教授在生前最后访谈中透露，他始终相信该理论具有“中间理论”的独特价值。如今，曾被视作理论毒药的幽灵粒子，正引领物理学家走向量子引力研究的新征程。

中文翻译：

古老量子引力“幽灵”理论迎来复兴

我们最熟悉的力，至今仍是最大的谜。物理学家已知名为光子的粒子洪流如何点亮我们的生活，也知道“胶子”粒子群如何束缚原子核。但若问及引力由何种粒子构成（倘若存在的话），又是何种粒子让我们的勺子坠向地面逗乐婴儿，他们便无言以对。用粒子理论解释引力如此困难，以致许多物理学家彻底放弃了这条路径。他们开始思考另一种可能：引力乃至整个现实，或许由微小弦或其他奇异物质构成。

然而在理论物理学的某个角落，粒子理论正悄然复苏。越来越多的物理学家开始运用典型粒子物理学方法——量子场论来研究引力。尽管这种理论长期被认为存在致命缺陷，但如今研究者发现其表现远超前辈预期。

“目前没有任何迹象表明应该抛弃量子场论，恰恰相反。”荷兰拉德堡德大学理论物理学家卢卡·博宁凡特表示，他的计算为这一古老理论提供了支撑。他指出，将标准量子场论应用于引力时，不仅会得到名为“二次引力”的独特理论，“更会获得全新预言”。

这些预言尚未经受检验。纯粹从理论角度看，二次引力具有令人不安的特性，仍使许多物理学家望而却步。

但二次引力的支持者并未因这些异常而退缩。相反，他们将这些特征视为量子场论中未被充分认知的潜在可能。例如在微观层面，结果或许偶尔会先于原因出现；二次引力中出现的负能量“幽灵”粒子，或许能安然存在于方程中而不会引发实验悖论。

博宁凡特认为，幽灵粒子可能正是“我们深入理解引力与量子场论时涌现的新客体”。

常数越多，难题越甚
自物理学家尝试将引力纳入量子场论（他们用以描述其他所有基本力的框架）起，这场联姻就注定坎坷。

量子场是弥漫于空间的涟漪状物质。场中的涟漪即是粒子。通过交换这些粒子涟漪流，物体间能够相互推拉产生作用力。例如电磁力就是通过电磁场中被称为光子的扰动传递的。

量子场论有个极不便的事实：场的行为取决于其可能支撑的每一道涟漪。而这些涟漪具有无穷无尽的形态与尺度。当物理学家最初创立量子场论并用以研究电子和光子时，他们的计算会得出无穷大结果——因为求和的每一项都试图涵盖永无止境的微小涟漪连续体。但无穷项求和根本得不出有效答案。

1940年代末，物理学家理查德·费曼、朱利安·施温格和朝永振一郎各自独立找到了解决之道，将这些无尽涟漪转化为明确答案，并因此荣膺诺贝尔奖。他们意识到，可以用两个实验室已测得的常数——电子质量与电荷——的净效应来重新表述计算中那些不可知且看似无穷的部分。此举固定了各项数值，此后物理学家便能预测电磁场的任何特性。

这种被称为重整化的技巧看似代数取巧。但在随后数十年间，物理学家逐渐理解了其有效性。重整化是通过模糊场中最微小的涟漪，仅保留其净效应来实现的。对电磁场而言，此法可行是因为小涟漪的影响有限：涟漪越小，对大尺度涟漪的影响就越弱。

然而引力的运作方式截然不同。

引力同样拥有自己的场：时空结构本身。阿尔伯特·爱因斯坦在广义相对论中将引力描述为物体沿时空结构曲线“坠落”的结果。引力场本身并非充满空间的涟漪物质，而是构成空间的涟漪物质。物理学家已探测到穿越此场的“引力波”。而这场中最微小的涟漪却带来了无穷麻烦。

当费曼与同事布莱斯·德威特尝试对引力进行重整化时，他们发现时空涟漪越小，其影响反而越大。这些涟漪以无数难以察觉的方式影响着更高层级的时空波动，无法仅用少数可测常数来概括。技巧至此失效。微小时空涟漪拒绝被模糊处理。

“所有人都为此困扰，”马萨诸塞大学阿默斯特分校量子场论专家约翰·多诺霍表示，“这正是量子广义相对论被视为难题的原因。”

二次引力的诞生
1970年代中期，当时还是布兰迪斯大学研究生的已故学者凯洛格·斯特尔发现了一种方法——也是唯一的方法——能够阻止早期“量子化”广义相对论尝试中无穷大结果的泛滥。

广义相对论可以写成一个代表时空曲率的单项方程。对此方程应用费曼与德威特的重整化程序，会得到一种粒子涟漪——引力子，其波动方式会不可避免地产生无穷大。

但斯特尔发现可以通过修改爱因斯坦方程，使时空更接近电磁场的特性：涟漪越小，其影响越弱。这样它们的整体效应就能用少数可测常数来概括，类似于电磁学中的电子电荷与质量。这种引力理论因包含两个与曲率平方相关的新项，被称为二次引力。它具有可重整性，与电磁学同样自洽。

“这给出了一个量子引力理论，”上月逝世的伦敦帝国理工学院教授斯特尔生前坦言，“但问题在于：你是否喜欢这个理论？”

大多数物理学家——包括斯特尔本人——并不满意。

“我当时就意识到并非所有细节都完美。”他在四月的一次访谈中如是说。

问题在于这种增强的时空结构现在能支撑三种涟漪。第一项代表常规引力子，但两个曲率平方项引入了两种新粒子。其中一种无害，被斯特尔称为“可爱的小标量粒子”；另一种却是怪物。

第三项中一个不受欢迎的负号引发了混乱。相关粒子具有负能量，时空结构实际上会通过创造它而获得能量。这意味着此类粒子会自发大量涌现，将空间本身搅动成能量不断攀升的炼狱。

更糟的是，涉及第三种粒子的事件可能具有负概率——这完全违背物理意义。

物理学家称此类粒子为幽灵，并称被幽灵纠缠的理论在数学上“病态”。

斯特尔1977年发表论文后不久，物理学家偶然发现了更健康的量子引力理论“超引力”。该理论通过假定每种已知基本粒子都与尚未发现的“超伴子”配对，解决了一系列难题。超引力立即吸引了包括斯特尔在内的理论物理学家关注，最终与弦理论融合主导该领域数十年。

带有幽灵粒子与不自洽缺陷的二次引力无力竞争。物理学家对其关注甚少，斯特尔的论文每年仅被引用10至20次。

幽灵复苏
然而该理论从未彻底消失。

理论学家们断断续续地回归研究。2010年代，随着弦理论未能实现早期研究者承诺的重大突破，且大型强子对撞机实验中未发现超伴子，人们对二次引力的兴趣重新燃起。

2014年，意大利物理学家阿尔贝托·萨尔维奥和亚历山德罗·斯特鲁米亚思考二次引力能否解决众人期待超伴子破解的难题。这个被称为层次性问题的问题在于：为何引力与其他三种基本力相比微弱得不可思议？为何其他力遵循某种标度，而引力却适用截然不同的特殊标度？萨尔维奥和斯特鲁米亚发现斯特尔理论中的两个额外粒子有助于分离这两种标度。这让他们开始怀疑幽灵粒子是否真是不可逾越的障碍。

几年后，比萨大学的达米亚诺·安塞尔米发现，研究者通过采用费曼量子事件描述规则的替代版本，可以规避幽灵理论遭遇的陷阱。“人们原以为已有定论，但事实并非如此。”他表示。

近期因对量子场论的贡献荣获J·J·樱井奖的多诺霍也开始研究这个被称为“病态”的理论。他与现任圣保罗大学的加布里埃尔·梅内泽斯合作发现，在简单情境中，幽灵粒子实际上不会引发物理学家担忧的灾难。它们极不稳定，往往在激发真空或显现负概率之前就已消失。真空保持稳定，概率总和始终维持100%——这是被称为幺正性的基本特性。

“我们有几个让我信服的案例，”多诺霍说，“其他人认为违反幺正性的情况似乎并未发生。”

为何幽灵只是潜伏而非肆虐？二次引力支持者提出了几种相互关联的解释。

萨尔维奥与多伦多大学荣休物理学家鲍勃·霍尔登分别注意到，可以通过调整概率计算中最终（且相当可疑）的步骤，确保概率恒为正。

多诺霍指出，即便不按安塞尔米建议的方式修改费曼规则，幽灵粒子也几乎不存在。它们仅瞬现于极短距离。这些瞬间需要付出代价，但并非稳定性或幺正性，而是通常严格遵循的因果顺序。那个负号允许幽灵粒子短暂时间倒流，影响原本无法触及的粒子。在这种图景下，我们体验的时间单向流动，实则是大量时间可塑的微观时刻的精密平均。

理解其他基本力都需要物理学家掌握重整化等新奇概念。因此研究者认为，或许幽灵粒子（以及处理它们所需的量子场论修正规则）正是解开引力之谜的钥匙。即便引力并非以此方式运作，物理学家仍认为严谨探索含幽灵负号的量子场论行为具有价值。“这其中确有深意。”研究其他引力理论的拉德堡德大学研究员本杰明·克诺尔表示。

要确定这些标准量子场论扩展版本的兼容性及其在更复杂情境中是否失效，仍需大量工作。但多数二次引力研究者已不再畏惧幽灵。“数学上它们现在能说通了。”博宁凡特表示。

不过其他理论家仍对这些修正能否解决所有潜在问题存疑。因果律与幺正性等物理学核心原则不容轻易改动。“他们很努力，”哥本哈根大学量子引力研究员阿莱西亚·普拉塔尼亚表示，“但我认为这仍是未解之谜。”

现实的层级
对二次引力研究者而言，驱幽灵的探索虽充满不确定性却值得投入。除了理论上的数学优势，斯特尔所说的“可爱小标量”粒子正是那种可能驱动宇宙大爆炸初期暴胀的粒子（及相关量子场）——许多宇宙学家认为这塑造了我们今日所见的宇宙。事实上，苏联物理学家阿列克谢·斯塔罗宾斯基于1980年运用二次引力首次构建了宇宙初期暴胀的理论。

宇宙暴胀应在时空中产生涟漪，在天空留下细微印记。尽管经过密集搜寻仍未发现这些印记，排除了若干暴胀模型。但安塞尔米等人近期研究指出，二次引力宇宙产生的涟漪可能微弱到现有望远镜无法探测。下一代天文台或许能捕获这些微弱波动。

“在我看来，斯塔罗宾斯基暴胀是唯一从量子场论角度能自洽的模型。”多诺霍表示。

凭借意外温顺的幽灵粒子、斯塔罗宾斯基暴胀模型日益流行以及其他量子引力理论的停滞不前，二次引力逐渐重获关注。斯特尔的原初论文近年每年获引超过150次。

若有朝一日幽灵被彻底驯服，斯塔罗宾斯基暴胀的时空涟漪被探测到，二次引力将如何揭示现实本质？观点各异。

弦理论、黑洞物理学等领域的线索使学界普遍认为，时空在亚微观尺度应解离为更奇异物质。但若二次引力终成引力终极理论，那么震颤的时空结构将无论放大多少倍始终存在。“我们谈论的是直至任意小尺度的真实连续描述，”霍尔登说，“时空永恒。”

这种可能性近期获得佐证。去年多诺霍与合作者发现关于二次引力中引力子碰撞的重要数学事实：碰撞越剧烈，引力越微弱，计算反而更易——这种现象称为渐近自由。他们的结果表明二次引力永不失效，可直抵现实最深层次。“这或许能使其成为终极理论，”多诺霍说，但他补充道，“我尚未确信这就是最终答案。”

另一种可能是：尽管具有可重整性与渐近自由，二次引力仍非引力的完整描述。

须知重整化相当于给世界加装滤镜，模糊其最微小的涟漪。可重整理论中，忽略这些细节不会显著改变整体图景——模糊图像已足够有效。不可重整理论则不然，模糊图像与清晰图像差异巨大——因每个无限小细节都举足轻重，故难以理想适用。因此如量子化广义相对论等不可重整理论，要求物理学家同时理解现实的所有层级。

二次引力的意外成功暗示，引力或许确实存在足够有效的模糊图像。在特定空间尺度之下，任何复杂细节——无论是弦、圈还是虚无——或许皆可忽略，仍能获得完全自洽的理论。若真如此，物理学家便可准确预测引力子碰撞与宇宙暴胀，而无须担忧最小尺度发生的真相。“它未必是终极理论，”多诺霍说，“但或许能成为现实中一个封闭自洽的层级。”

斯特尔亦持此见。他虽未亲自回归自己的理论，但一直远观其近期复兴（他称之为“令人欣慰”）。他逐渐意识到这个理论终究具有价值。“我目前的看法是，这是一个可能的链接，”他说，“一种中间态。”

英文来源：

Old ‘Ghost’ Theory of Quantum Gravity Makes a Comeback
Introduction
The force we experience most intimately remains the most mysterious. Physicists understand how vast migrations of particles called photons light up our homes, and how swarms of “gluon” particles hold together the cores of our atoms. But they can’t say what gravity particles, if any, delight us as babies by enabling our spoons to plummet to the floor. The force of gravity has proved so difficult to account for in terms of particles that many physicists have abandoned that approach altogether. They consider the possibility that gravity — and with it, reality as a whole — might instead be made of tiny strings or other strange things.
But in one corner of the theoretical-physics world, the particle approach is staging a comeback. A growing band of physicists has been using the typical approach to particle physics, known as quantum field theory, for gravity. Although this use of the theory was long considered fatally flawed, these physicists are now finding that it works far better than their predecessors expected.
“So far there is no hint telling us that we should throw quantum field theory away; actually, it’s the opposite,” said Luca Buoninfante, a theoretical physicist at Radboud University in the Netherlands whose calculations have helped shore up the old theory. When you apply the standard quantum field theory to gravity, you don’t just get a unique theory called quadratic gravity, he said. “You also get new predictions.”
Those predictions have not yet been tested. And on purely theoretical grounds, quadratic gravity has eerie features that still spook many physicists.
But quadratic gravity enthusiasts aren’t put off by its abnormalities. On the contrary, they view these features as previously unappreciated possibilities that may be permitted by quantum field theory. Perhaps effects occasionally sneak ahead of their causes at the microscopic level, for instance. And perhaps negative-energy “ghost” particles that arise in quadratic gravity can exist safely in the equations without creating paradoxes in experiments.
Ghosts, Buoninfante said, may be “new objects that appear when we try to understand gravity and quantum field theory at a deeper level.”
More Constants, More Problems
From the moment physicists tried to fit gravity into quantum field theory (the framework they use to describe all the other fundamental forces), it was obvious the union was going to be a rocky one.
Quantum fields are rippling substances that suffuse space. A ripple in a quantum field is a particle. By exchanging streams of these particle-ripples, one object can push or pull on another, exerting a force. The electromagnetic force, for instance, is conveyed through disturbances in the electromagnetic field that we call photons.
A deeply inconvenient truth of quantum field theory is that what a field does depends on each and every one of the ripples it can conceivably support. And those ripples come in an unending number of shapes and sizes. When physicists first invented quantum field theory and tried to use it to ask questions about electrons and photons, their calculations went infinite because each term in a sum tried to account for a never-ending continuum of ever-smaller ripples. But a sum of infinite terms was no answer at all.
In the late 1940s, the physicists Richard Feynman, Julian Schwinger and Sin-Itiro Tomonaga independently hit on a workaround that would turn those endless ripples into clear answers, earning themselves a Nobel Prize. They realized that they could reexpress the unknowable, infinite-seeming parts of their calculations in terms of their net effect on two known constants that had already been measured in the lab: the electron’s mass and charge. Doing so fixed the values of the terms, after which physicists could predict anything they liked about the electromagnetic field.
This trick, known as renormalization, seemed like an algebraic hack. But over the following decades, physicists came to understand why it worked. Renormalization was a way of blurring out the smallest ripples in a field and including only their net effects. In the case of the electromagnetic field, this works because the impact of the small ripples is limited; the smaller the ripples get, the less they influence the larger ripples.
Gravity, however, works differently.
Gravity too has a field: the fabric of space-time itself. Albert Einstein, in his general theory of relativity, described gravity as a consequence of objects “falling” along curves in this space-time fabric. This gravitational field is not a rippling thing that fills space, per se, but rather a rippling thing that is space. Physicists have detected “gravitational waves” traveling through this field. And the tiniest ripples in this field cause no end of trouble.
When Feynman and a colleague by the name of Bryce DeWitt tried to renormalize gravity, they found that the tinier the space-time ripples, the more they matter. They influence the rippling of space-time at higher levels in innumerable subtle ways that can’t be summed up only in terms of a few measurable constants. The trick failed. The tiny space-time ripples refused to be blurred out.
“Everyone was concerned about this,” said John Donoghue, an expert in quantum field theory at the University of Massachusetts, Amherst. “This is the reason that quantum general relativity was considered a problem.”
Birth of Quadratic Gravity
In the mid-1970s, the late Kellogg Stelle, then a graduate student at Brandeis University, saw that there was a way — and only one way — to stop the inundation of infinities that had plagued earlier attempts to “quantize” general relativity.
General relativity can be written as an equation that has a single term representing space-time’s curvature. Apply Feynman and DeWitt’s renormalization procedure to this equation and you get one type of particle-ripple, the graviton, rippling in unignorably infinite ways.
But Stelle figured out that he could modify Einstein’s equation so that space-time more closely resembled the electromagnetic field, with ripples that became less significant as they got smaller. Their overall effects could then be captured in just a few measurable constants, analogous to the electron’s charge and mass in electromagnetism. This theory of gravity, which came to be known as quadratic gravity because it contained two new terms related to the square of the curvature, was renormalizable. It made just as much sense as electromagnetism.
“That gives you a — indefinite article — quantum theory of gravity,” said Stelle, who was a professor at Imperial College London until his death last month. “Then of course the question is: Do you like it?”
Most physicists, Stelle among them, did not.
“I was also aware that not everything was going to be hunky-dory about this,” he said during an interview in April.
The problem was that this enhanced fabric of space-time could now host three types of ripples. The first term represents the normal gravitons. But the two curvature-squared terms bring two new particles into the picture. One is inoffensive, what Stelle called a “sweet little scalar” particle. But the other is a ghoul.
An unwelcome minus sign stemming from the third term unleashes chaos. The associated particle has negative energy, so the space-time fabric actually gains energy by creating it. This means that more and more of such a particle will spontaneously appear, whipping space itself into an increasingly energetic inferno.
Worse, events involving this third particle may have a negative probability of taking place — a meaningless proposition.
Physicists call such particles ghosts and say that theories haunted by ghosts are “sick” — mathematically inconsistent.
Shortly after Stelle published his research in 1977, physicists stumbled across a healthier quantum gravity theory called supergravity. It solved a handful of problems by positing that each known elementary particle teams up with an as-yet-undiscovered “superpartner” particle. Supergravity immediately captured the attention of theoretical physicists, including Stelle. The theory would eventually merge with string theory and dominate the field for decades.
Quadratic gravity, with its ghosts and inconsistencies, couldn’t compete. Physicists paid it little attention, citing Stelle’s paper just 10 to 20 times per year.
Ghostly Revival
The theory never completely faded away, however.
Theorists returned to it here and there. Interest rose in the 2010s as string theory failed to deliver the spectacular breakthroughs its early practitioners had promised, and superpartners failed to materialize in experiments at the Large Hadron Collider.
In 2014, the Italian physicists Alberto Salvio and Alessandro Strumia wondered whether quadratic gravity could solve a puzzle that many had expected the superpartners to address. The puzzle, known as the hierarchy problem, asks why gravity seems impossibly weak compared to the other three fundamental forces. Why is there one “scale” for those forces, and a special, dramatically different one for gravity? Salvio and Strumia found that the two extra particles from Stelle’s theory could help push the two scales apart. That got them wondering whether the ghost was truly a deal-breaker.
A few years later, Damiano Anselmi of the University of Pisa found that researchers can avoid the pitfalls encountered by theories with ghosts by using alternative versions of the rules Feynman laid out for describing quantum events. “You have the impression that the last word had been said. It was just not true,” he said.
Donoghue, who recently won the prestigious J.J. Sakurai Prize for his contributions to quantum field theory, also started studying the allegedly sick theory. Working with Gabriel Menezes, now of the University of São Paulo in Brazil, he found that in simple scenarios, the ghost particles don’t actually wreak the havoc physicists feared. They are so unstable, they tend to vanish before they have time to fire up the vacuum or manifest negative probabilities. The vacuum stayed calm, and probabilities continued to add up to 100% — an essential property known as unitarity.
“We have a few examples, which are what made me believe it,” Donoghue said. “Things everyone else would have said violated unitary don’t seem to.”
So why does the ghost seem to silently lurk, rather than actively haunting Stelle’s theory? Quadratic gravity enthusiasts have hit on a couple of overlapping ideas.
Salvio and Bob Holdom, a physicist emeritus at the University of Toronto, have independently noticed that it’s possible to tweak the final (and rather suspect) step in the calculation of probabilities in a way that guarantees they’ll always stay positive.
And Donoghue points out that even if you don’t change Feynman’s rules in the manner proposed by Anselmi, ghosts barely exist. They show up only fleetingly, over short distances. In those instants there is a heavy price to be paid, but it isn’t stability or unitarity. Rather, it’s the normally rigid ordering of cause and effect. That minus sign allows ghost particles to briefly skip backward in time, where they can influence particles that they otherwise couldn’t. In this picture, the inexorable forward flow of time that we experience would emerge as a delicate average over lots of temporally squishy micro-moments.
Understanding each of the other fundamental forces required physicists to master strange new concepts like renormalization. So, the researchers argue, perhaps ghosts (and the modified rules of quantum field theory that are needed to deal with them) are the key that will unlock gravity. And even if gravity doesn’t work this way, physicists see value in rigorously exploring the behavior of quantum field theories with ghostly minus signs. “There is something to understand there,” said Benjamin Knorr, a researcher at Radboud University who works on other gravity theories.
Much work remains to determine how compatible these proposed extensions of standard quantum field theory are with each other, and whether they will fail in more demanding situations. But for the most part, quadratic gravity researchers have stopped fearing ghosts. “Mathematically, they make sense now,” Buoninfante said.
Other theorists, however, still harbor qualms as to whether these fixes really address all the potential problems. One does not simply tinker with critical tenets of physics like causality and unitarity. “They’re working hard,” said Alessia Platania, a quantum gravity researcher at the University of Copenhagen. “But I would say it’s still open as a question.”
Layers of Reality
For quadratic gravity researchers, the uncertain work of ghost-busting is worthwhile. On top of the theory’s mathematical advantages, Stelle’s “sweet little scalar” is exactly the sort of particle (and associated quantum field) that could have driven an explosive expansion of the cosmos during the Big Bang, which many cosmologists believe set up the universe we see today. In fact, the Soviet physicist Alexei Starobinsky used quadratic gravity to formulate the first theory of that initial growth spurt, termed cosmic inflation, in 1980.
Cosmic inflation should have set off ripples in space-time that left subtle imprints on the sky. These haven’t been seen despite intense searching, ruling out several models of inflation. Recent work from Anselmi and others, however, suggests that a universe with quadratic gravity should produce ripples too small for current telescopes to detect. Next-generation observatories may be able to pick up these weak waves.
“Starobinsky inflation is the only remaining one that makes sense from a quantum field theory point of view to my taste,” Donoghue said.
Through a combination of unexpectedly friendly ghosts, the growing popularity of Starobinsky inflation, and other quantum gravity theories languishing in the doldrums, quadratic gravity has grown in popularity. Stelle’s original paper has recently been accruing more than 150 citations each year.
If the ghosts can be fully vanquished and the space-time ripples from Starobinsky inflation are detected someday, what would quadratic gravity imply about reality? Opinions vary.
Clues from string theory, black hole physics and other places have led to a widespread belief that space-time should unravel into stranger stuff at some submicroscopic scale. But if quadratic gravity turns out to be the ultimate theory of gravity, then the quivering space-time fabric could persist no matter how far you zoom in. “We’re talking about a true continuum description all the way to arbitrarily small scales,” Holdom said. “Space-time forever.”
That possibility recently received a boost. Last year, Donoghue and collaborators discovered an important mathematical fact about how gravitons collide in quadratic gravity. As collisions get more intense, gravity gets weaker, making calculations easier — a phenomenon known as asymptotic freedom. Their result suggests that quadratic gravity never breaks down and could take you all the way to the deepest levels of reality. “This could allow it to be the final theory,” Donoghue said. However, he added, “I’m not convinced that this is the final theory.”
The alternative is that quadratic gravity, despite being renormalizable and asymptotically free, still isn’t a complete account of gravity.
Recall that renormalization amounts to a filter over the world that blurs out its tiniest ripples. Renormalizable theories are those in which glossing over those features doesn’t change the overall picture much. The blurry picture works well. Non-renormalizable theories are those where the blurry picture looks quite different from the sharp picture. It doesn’t work as well as you’d like because every infinitesimal detail matters. So non-renormalizable theories, like quantized general relativity, challenge physicists to understand all of reality’s layers at once.
The surprising successes of quadratic gravity hint that gravity may yet turn out to have a blurry picture that works well enough after all. Below a certain spatial scale, it may be that any complicated details — whether those are strings, loops or nothing at all — can be ignored, and you’ll still get a fully consistent theory. If that’s the case, physicists can accurately predict how gravitons collide and how the universe inflated without worrying about what’s truly going on at the smallest scales. “It may or may not be the ultimate theory,” Donoghue said. But maybe “it becomes a closed, self-consistent layer of reality.”
Stelle shared that view. He never returned to his theory himself, keeping tabs on its recent revival (which he called “agreeable”) from afar. But he came to suspect that it had some value after all. “My own reaction to it right now is, it’s a possible link,” he said, “a kind of intermediate regime.”