5 Jalan Merumuskan Teori Gravitasi Kuantum

Bagaimana Gravitasi Kuantum Ditemukan

Teori gravitasi kuantum yang mencoba menggabungkan teori relativitas umum dengan teori medan kuantum masih terus diburu oleh para saintis dan para fisikawan. Walaupun masing-masing teori, baik teori relativitas umum maupun teori medan kuantum adalah teori yang kuat karena teruji konsisten pada banyak sekali eksperimen, menggabungkan keduanya sangat sulit.

Perburuan ini sangat penting dalam fisika karena diharapkan bisa memberikan gambaran kondisi pada saat Big Bang, bagian terdalam dari Black Hole, dan menyatukan empat gaya fundamental di alam.

Berikut ini berapa kemungkinan jalan atau tarekat yang membuka ditemukannya teori gravitasi kuantum tersebut.

1. Menghubungkan osilasi gelombang gravitasi dengan osilasi medan kuantum.
2. Mentranformasi geometri riemannian ke dalam skala panjang planck dan waktu planck.
3. Menghubungkan frekuensi pada medan kuantum dengan massa pada relativitas umum.
4. Menghubungkan antara lubang cacing dengan prinsip entanglement.
5. Menghubungkan koordinat ruang dan waktu pada persamaan Dirac pada teori medan kuantum dengan koordinat ruang dan waktu Einstein pada relativitas umum.

Yuk kita lihat satu per satu.

Osilasi Gelombang Gravitasi - Medan Kuantum

Tensor energi momentum pada persamaan medan Einstein memberitahu kita bahwa energi, massa, dan percepatan memengaruhi kelengkungan ruang-waktu. Perubahan pada ketiganya akan membentuk osilasi pada ruang-waktu. Osilasi itu disebut sebagai gelombang gravitasi.

Osilasi gelombang gravitasi yang besar biasanya disebabkan oleh tabrakan lubang hitam, tabrakan galaksi, tabrakan bintang neutron, sistem bintang biner, dan supernova.

Dalam skalanya yang sangat kecil, gelombang gravitasi bisa disebabkan oleh gerak rotasi dan revolusi benda-benda angkasa. Ini artinya ruang-waktu terus menerus dalam keadaan berosilasi.

Hal yang sama juga terjadi pada medan kuantum. Medan kuantum senantiasa berosilasi akibat proses nihilisasi matter-antimatter virtual pada tingkat energi ground state.

Menemukan formalisme matematis yang bisa menghubungkan kedua osilasi ini bisa membuka jalan bagi teori gravitasi kuantum. Bisa jadi ini, yaitu medan kuantum dan relativitas umum, adalah manifestasi berbeda dari fenomena yang sama. Seperti halnya magnet dan listrik yang sebenarnya merupakan satu fenomena tunggal yaitu elektromagnetik. Persamaan Maxwell telah berhasil menyatukan kedua manifestasi magnet dan listrik tersebut.

Karena benda-benda angkasa terus menerus bergerak, maka ruang-waktu senantiasa berosilasi.  Kita cuma butuh detektor yang lebih sensitif untuk bisa menangkap gelombang gravitasi yang lemah. Ketika detektor yang dimaksud ditemukan, kita akan melihat osilasi serta efek lanjutannya berupa kerutan dan interferensi antar gelombang gravitasi ini adalah sifat utama dari ruang-waktu. Fenomena ini identik dengan apa yang terjadi pada medan kuantum. Dengan demikian, transformasi matematis antara relativitas umum dengan medan kuantum bisa dilakukan. 

Geometri Riemannian Pada Skala Planck

Geometri Riemannian digunakan dalam relativitas umum untuk menggambarkan kelengkungan ruang-waktu akibat dari tensor energi momentum. Ruang-waktu dalam skala Planck terbentuk dari panjang Planck dan waktu Planck.

Masalah utamanya, selain belum adanya formalisme matematis yang diterima secara umumkan antara ruang dan waktu dalam skala Planck dengan ruang-waktu pada relativitas umum, adalah panjang planck dan waktu planck tidak bisa dijangkau oleh alat eksperimen kita saat ini. Jembatan matematis ini juga menghadapi paradoks eksistensi energi tinggi pada skala Planck. Suatu tingkat energi yang setara dengan seluruh energi semesta yang ada sekarang.

Dalam skala Planck, ukuran ruang-waktu yang sangat kecil dengan tingkat energi yang sangat tinggi, membawa konsekuensi yang tidak mudah seperti lenyapnya ruang-waktu seketika setelah terjadi Big Bang.

Sebuah makalah berjudul On the Same Origin of Quantum Physics and General Relativity from Riemannian Geometry and Planck Scale Formalism yang terbit pada Jurnal Astroparticle Physics 22 Agustus 2024 mencoba menemukan jembatan matematis antara Geometri Riemannian dengan Skala Planck. 

Makalah tersebut dibahas oleh Sabine Hossenfender pada 10 September dan kemudian ditertawakannya pada akun YouTube nya. Bahkan Sabine meremas dan melemparkan makalah itu ke kamera. Ini artinya rumusan matematisnya salah total. Tapi itu tidak menunjukkan konsepnya salah.

Menemukan formalisme matematis untuk menerapkan skala Planck pada Geometri Riemannian yang bisa diterima komunitas ilmiah secara luas masih merupakan tantangan yang mengasyikkan.

Osilasi ruang-waktu akan memaksa kita memikirkan ulang konsep ruang waktu yang smooth, continous, homogen, dan isotropik. Pada akhirnya sejumlah besar gelombang gravitasi akan membentuk kerutan dan interferensi di mana-mana dengan pusat-pusat materi dan energi yang tidak merata di seluruh kain ruang-waktu yang ada.

Relasi Frekuensi - Massa

Radioaktif decay, baik itu alfa decay, beta decay, maupun gamma decay menghasilkan foton. Pada peristiwa radioaktif decay, massa diubah menjadi foton yang sepenuhnya energi.  Sementara tabrakan antara foton dengan foton menghasilkan pasangan materi-anti materi.  Ini membawa kepada hubungan antara energi foton dengan kesetaraan energi massa.  

Jika energi foton adalah E = hf, sedangkan dalam level materi berlaku kesetaraan energi-massa E = mc^2, maka jika saja kita menemukan formalisme matematis yang memungkinkan hf = mc^2, gravitasi kuantum bisa lebih mudah dirumuskan.

Baik h maupun c adalah konstanta, maka lebih lanjut f = m yaitu di mana frekuensi bisa setara dan terhubung dengan massa.

Di level kuantum, foton dihasilkan oleh electromagnetic field dan radioaktif decay, sementara di level relativitas umum foton dihasilkan dari supernova. Foton juga membawa informasi semesta dan mengalami pergeseran merah searah dengan percepatan perluasan semesta.

Ketika kita dapat menemukan formalisme matematis yang menghasilkan kesetaraan f = m, maka kita bisa memproduksi gambaran dan narasi yang fasih tentang apa yang terjadi pada saat Big Bang dan titik terdalam dari lubang hitam.

Bisa jadi di titik terdalam lubang hitam dan Big Bang segala sesuatunya ada dalam bentuk frekuensi. Pada titik ini teori string menemukan momentum dukungannya. Segala sesuatunya adalah vibrasi dan osilasi frekuensi yang berbeda. Setiap frekuensi mewakili medan kuantum dan partikel elementer yang berbeda. 

Formalisme f = m sebagai turunan dari foton menjadi penting karena sifat foton yang secara luas berinteraksi dengan semua partikel elementer dan gaya fundamental yang ada.

Foton berinteraksi depan gravitasi melalui gravitational lensing, kemudian interaksi foton dengan boson lain seperti boson w-z adalah kesatuan gaya yang sama dalam bentuk gaya electroweak.

Sementara interaksi foton dengan gluon pada inti atom menghasilkan pasangan partikel quark-anti quark.

Semua interaksi ini memperkuat hipotesis bahwa foton bisa menjelaskan dan menghubungkan interaksi di level medan kuantum dan relativitas umum.

ER = EPR

Diduga lubang cacing dan entanglement adalah dua manifestasi yang berbeda dari satu fenomena yang sama. Di mana entanglement terhubung oleh lubang cacing, dalam arti informasi kuantum disalurkan melalui lubang cacing.

ER dipersamakan dengan EPR karena keduanya memiliki persamaan dalam hal dua kondisi yang saling terhubung di mana salah satu kondisi akan memengaruhi kondisi lainnya sehingga kedua kondisi itu tidak dapat dipisahkan dan, lebih lanjut memungkinkan transfer informasi dengan cepat.

Konjungtur dan hipotesis ini telah diusulkan oleh Juan Maldacena dan Leonard Susskind pada tahun 2015. Hipotesis mereka masih diragukan karena framework dan formalisme matematisnya dianggap lemah dan tidak didukung oleh eksperimen.

Walaupun begitu, tampaknya yang lemah itu adalah formalisme matematisnya, dan bukan konsep dasarnya yang memungkinkan medan kuantum bisa dipertemukan dengan relativitas umum secara geometri.

Pada tahun 2022 telah berhasil dibuat di laboratorium Google oleh Maria Spiropulu suatu lubang cacing virtual yang memadukan sifat entanglement dengan prinsip holograpik.

Lubang cacing virtual ini berhasil membuktikan transfer energi secara cepat pada dua kondisi yang terpisah. Pembuatan lubang cacing ini mengambil pelajaran dari hipotesis ER =EPR. JIKA ER tidak sama dengan EPR, lubang cacing virtual itu bisa jadi tidak akan terbentuk.

Ketika kerangka dan formalisasi matematis yang kuat dalam konteks ER = EPR ditemukan, maka rekonsiliasi antara medan kuantum dengan relativitas umum menjadi nyata.

Ruang-waktu Dirac dan Einstein

Ruang-waktu dalam persamaan Dirac bersifat datar, sedangkan ruang-waktu dalam relativitas umum bersifat melengkung dengan tingkat lengkungan tergantung besaran massa dan energi.

Jika dimungkinkan adanya formalisme matematis yang memungkinkan mengubah variabel ruang-waktu datar persamaan Dirac menjadi ruang-waktu melengkung, maka hambatan dalam merumuskan gravitasi kuantum bisa diatasi.

Mengapa variabel ruang-waktu Dirac yang harus diubah dan bukan ruang-waktu Einstein? Ini karena ruang-waktu datar dalam persamaan Dirac tidak bisa menampung perubahan dalam tensor energi momentum.

Eksistensi variabel massa dalam persamaan Dirac dalam perspektif ruang-waktu relativitas umum memungkinkan ruang-waktu yang melengkung diterapkan dalam persamaan Dirac.

Penggunaan massa, matrik gamma, dan koordinat ruang-waktu menjadikan persamaan Dirac mirip dengan massa, tensor energi momentum, dan geometri riemannian dalam relativitas umum. Ini membuka kemungkinan transformasi di antara keduanya.

Uji Hipotesis 

Mari kita uji 5 hipotesis kita di atas dengan menggunakan AI ChatGPT OpenAI. 

Dirac equation consists of gamma matrix, spatial ordinate, and mass in which identical with mass, ricci geometri, and tensor energy momentum, can we transform them each other?

The Dirac equation, when reformulated in the context of general relativity, does indeed link some of the structures you're asking about, but a direct transformation of the gamma matrices into mass, Ricci geometry, or energy-momentum tensors is not straightforward. However, let's break this down:

Dirac Equation in Flat Spacetime:
The Dirac equation in flat spacetime is given by:
(i\gamma^\mu \partial_\mu - m)\psi = 0
Gamma are the Dirac gamma matrices, which obey the Clifford algebra .
m is the mass of the fermion.
psi is the spinor field.
Mm is the Minkowski metric for flat spacetime.

Dirac Equation in Curved Spacetime:

When we generalize this to curved spacetime (i.e., include Ricci geometry), we replace partial derivatives with covariant derivatives and introduce vierbeins (or tetrads) to link the local flat spacetime (where the gamma matrices act) to the curved spacetime. The equation becomes:
(i\gamma^\mu e_\mu^a D_a - m)\psi = 0
a are the vierbein (or tetrad) fields, which relate local Minkowski space to curved spacetime.
D is the covariant derivative that now includes the spin connection, encoding how spinors are parallel transported in curved spacetime.

Ricci Tensor and Energy-Momentum Tensor:

The Einstein field equations relate the Ricci tensor  and the energy-momentum tensor  as:
R_{\mu\nu} - \frac{1}{2}g_{\mu\nu}R = 8\pi G T_{\mu\nu}
R is the Ricci curvature tensor, describing spacetime geometry.
T is the energy-momentum tensor, describing the distribution of matter and energy.
g is the metric of the curved spacetime.
Connections and Transformations:

The gamma matrices themselves cannot directly transform into the Ricci tensor or the energy-momentum tensor, as they belong to different algebraic structures. Gamma matrices are part of the Clifford algebra that operates in spinor space, while the Ricci tensor is a geometric object and the energy-momentum tensor is a physical object related to matter and energy.

However, when the Dirac equation is placed in curved spacetime, vierbeins (or tetrads) act as intermediaries between the local spinor space (with gamma matrices) and the curved geometry (with Ricci tensors).

The mass  remains a scalar in both cases, but it interacts with the geometry in terms of how the Dirac spinor couples to spacetime curvature.

In spinor-tensor theories and certain quantum gravity theories, these objects interact, and the spin connection (derived from the curvature of spacetime) influences the covariant derivative of the spinor field, linking the geometry to the behavior of fermions.

Conclusion:

While you cannot directly transform gamma matrices into the Ricci tensor or energy-momentum tensor, they do interact in the framework of curved spacetime. The vierbein formalism and the spin connection allow gamma matrices (acting in spinor space) to be consistently integrated with the geometry described by the Ricci tensor and the dynamics described by the energy-momentum tensor.

Can quantum gravity by making formalism of transformation among Dirac equation and general relativity equation?

In principle, quantum gravity aims to unify the quantum behavior of matter (as described by quantum field theory, including the Dirac equation for fermions) with the classical description of spacetime geometry provided by general relativity (via Einstein's equations). 

However, directly transforming the Dirac equation into the Einstein field equations (or vice versa) is not a straightforward task, as these equations describe different types of physical phenomena—matter and spacetime—though they interact closely.

Key Challenges in Unifying Dirac and General Relativity:

1. Different Mathematical Structures:
Dirac equation deals with fermions (spin-1/2 particles) and is written in terms of spinors, which are representations of the Lorentz group in local flat (Minkowski) spacetime.

General relativity describes the geometry of spacetime using the metric tensor and related curvature quantities (Ricci tensor, Einstein tensor, etc.), which operate in a very different mathematical framework (differential geometry and tensors).

Directly transforming one into the other would involve translating spinor fields and their dynamics into a geometrical language, which is not naturally built into general relativity.

2. Quantum vs. Classical Frameworks:
General relativity is fundamentally a classical theory—spacetime is treated as a smooth manifold described by Einstein's equations.
The Dirac equation belongs to quantum field theory and describes quantum particles, taking into account the principles of quantum mechanics such as superposition, uncertainty, and the discrete nature of energy levels.

Quantum gravity attempts to address how quantum matter (fermions, bosons) interacts with a quantized version of spacetime geometry.

Approaches Toward Quantum Gravity that Relate Dirac and Einstein Equations:

1. Coupling Dirac Equation to Curved Spacetime: The first step in linking quantum matter (Dirac spinors) with general relativity is placing the Dirac equation in curved spacetime. This is done by introducing vierbeins (tetrads) and a spin connection to express how spinor fields interact with the curvature of spacetime. The equation becomes:
(i\gamma^\mu e_\mu^a D_a - m)\psi = 0

2. Einstein-Cartan Theory: This theory is an extension of general relativity that includes torsion, which is related to the intrinsic spin of matter fields. In Einstein-Cartan theory:
The spin of fermions (from the Dirac equation) contributes to torsion in spacetime.
The torsion, in turn, affects the behavior of fermions. However, the metric and torsion are still classical fields, so this isn't a fully quantum theory of gravity, but it shows how spinor fields and spacetime can be linked.

3. Quantum Field Theory on Curved Spacetime: In this approach, quantum fields (like the Dirac field) are placed on a classical curved spacetime. The Dirac equation is modified to account for curvature, and quantum field theory techniques are applied. This allows for calculations such as Hawking radiation near black holes, but spacetime itself is not quantized.

4. Loop Quantum Gravity (LQG): Loop quantum gravity is a non-perturbative approach to quantizing spacetime itself, describing it as a network of discrete loops of quantum geometry. Fermions (via the Dirac equation) can be incorporated into this framework. Here, spacetime is fundamentally discrete at the smallest scales, and the interactions between fermions and spacetime are studied through this quantum-geometrical framework. While promising, LQG is still incomplete, and the full interaction between quantum matter and quantum geometry is an ongoing area of research.

5. String Theory: String theory aims to unify gravity with quantum mechanics by postulating that all particles, including fermions (from the Dirac equation) and gravitons (from general relativity), are different vibrational modes of fundamental strings. In this framework, the Dirac equation and Einstein's equations can both be seen as effective low-energy approximations of a more fundamental theory. Spacetime is also quantized in this approach, but the formalism is vastly different from that of conventional general relativity or quantum field theory.

6. Path Integral Formulation and Effective Field Theory: In some quantum gravity approaches (like in effective field theory), one might integrate out the Dirac fields (in a path integral approach) to derive an effective action for gravity. This allows a more formal way to see how quantum matter (described by the Dirac equation) could influence the dynamics of spacetime geometry (described by the Einstein field equations). 

However, this is more of an effective, approximate method rather than a full theory of quantum gravity.

Can We Transform Between Dirac and Einstein Equations?

A direct transformation between the Dirac equation and the Einstein field equations is not possible in the traditional sense because:
The Dirac equation describes fermionic quantum fields, while The Einstein field equations describe the classical curvature of spacetime.

However, quantum gravity aims to develop a formalism where both quantum matter and quantum spacetime can be treated consistently. In this sense, one might say that the coupling between Dirac fields and spacetime geometry (via tetrads, spin connection, etc.) allows us to study their interaction, but not a one-to-one transformation.

In quantum gravity frameworks like string theory or loop quantum gravity, both the spacetime geometry and the fermions are quantized, and their dynamics might be described by a unified formalism. However, this unification is still an area of active research, and no complete theory has yet emerged.

Conclusion:

While we cannot directly transform the Dirac equation into the Einstein field equations, quantum gravity approaches (such as loop quantum gravity, string theory, and others) attempt to describe a unified framework where quantum matter and quantum spacetime coexist and interact. These approaches seek to generalize both the Dirac and Einstein equations into a more fundamental quantum theory that captures the behavior of both matter and geometry at the smallest scales.

When mass movement in fabric of spacetime continum trigger ripple spacetime time, so that spacetime oscilate like quantum field, is it optional way to reconcile quantum field theory and general relativity theory?

Yes, the idea that mass moving through spacetime triggers ripples (gravitational waves) and that spacetime itself oscillates similarly to quantum fields is one potential pathway toward reconciling quantum field theory (QFT) with general relativity (GR). In essence, this refers to the broader concept of quantum gravity—a theory that attempts to describe gravity, not as a purely classical field as in general relativity, but as a quantized entity similar to other forces in nature.

Let's break down this idea and how it might help reconcile QFT and GR.

1. Spacetime Ripples and Gravitational Waves (General Relativity):
In classical general relativity, mass and energy distort the "fabric" of spacetime, and changes in the distribution of mass or energy can produce gravitational waves, ripples that propagate through spacetime itself. These were predicted by Einstein and confirmed by experiments such as LIGO's detection of gravitational waves from merging black holes.

Gravitational waves are analogous to ripples in spacetime, but in general relativity, spacetime is treated as a smooth, continuous entity, and these ripples are part of classical physics. Gravitational waves provide a direct link between the movement of mass in spacetime and oscillations in spacetime geometry, but this is still within the framework of a classical field.

2. Spacetime as a Quantum Field (Quantum Gravity):
To reconcile this with quantum field theory, one would need to quantize spacetime itself, treating it as a field that can exhibit quantum behavior, just like the electromagnetic field, for example. In quantum field theory, forces are mediated by particles (like photons for the electromagnetic force), and the idea would be to similarly quantize the gravitational field.

In quantum gravity, spacetime is no longer viewed as a smooth continuum but instead is expected to have a quantum structure at extremely small scales (the Planck scale). Just like the quantum fields that govern the other forces, spacetime would be subject to quantum fluctuations.

The ripples in spacetime (gravitational waves) would then be described by quantized excitations of the gravitational field, which are referred to as gravitons. These gravitons would be the hypothetical quantum particles that mediate gravitational interactions in a way analogous to how photons mediate electromagnetic interactions in QFT.

3. Spacetime Oscillations and Reconciliation with Quantum Field Theory:
If spacetime can oscillate like a quantum field, this opens up possibilities for unifying general relativity and quantum field theory in a single framework. This is the goal of quantum gravity. Some of the key approaches are:

(a) Perturbative Quantum Gravity:
This approach involves treating the gravitational field as a quantum field, similar to the other fields in QFT. In this context, gravitational waves are viewed as coherent states of gravitons (the quantized version of spacetime ripples). This method works well at low energies, but it runs into significant problems at higher energies, where the theory becomes non-renormalizable, meaning it generates infinite quantities that cannot be easily managed.

(b) String Theory:
String theory offers a different way to think about reconciling general relativity with quantum field theory. Instead of particles (like gravitons) being point-like, in string theory, they are seen as vibrating strings. These strings can represent both matter particles (like electrons) and force carriers (like photons and gravitons). In this framework, spacetime itself can emerge from the dynamics of strings, and gravitational interactions are described by the exchange of closed strings (which manifest as gravitons). String theory, therefore, attempts to quantize spacetime oscillations and mass-induced ripples within a consistent quantum framework.

(c) Loop Quantum Gravity (LQG):
LQG takes a more direct approach by trying to quantize spacetime itself. In this theory, spacetime is thought to be made up of discrete units, called spin networks. The ripples in spacetime caused by mass moving through it would correspond to changes in the structure of this quantum spacetime fabric. In this picture, spacetime is not continuous but made of small, quantized loops, and the ripples (like gravitational waves) are quantum phenomena.

4. Holographic Principle and Quantum Gravity:
Another approach that relates to spacetime oscillations and quantum fields is the holographic principle, which suggests that the description of a volume of spacetime can be encoded on its boundary. This concept comes from black hole physics, where the information of the black hole's interior seems to be encoded on its event horizon. 

The holographic principle hints at a deep connection between quantum field theory (QFT on the boundary) and the geometry of spacetime (general relativity in the bulk), and this could be another way to reconcile these two frameworks.

5. Effective Field Theories and Gravitational Waves:
In practical terms, gravitational waves themselves can be understood using effective field theory (EFT), a framework in which general relativity is seen as an approximation that breaks down at very high energies. Gravitational waves, which are oscillations in spacetime curvature, can be treated as low-energy phenomena. An effective quantum description of spacetime might be valid up to a certain scale (just as classical mechanics is an approximation of quantum mechanics at large scales), and at higher energies, a quantum theory of gravity would take over.

Conclusion: 

A Possible Path to Reconciliation?

The idea that spacetime oscillates like a quantum field is indeed a promising avenue to reconcile quantum field theory and general relativity. 

Quantizing spacetime (so that it exhibits quantum behaviors such as superposition and entanglement) would allow us to describe gravitational waves and other aspects of general relativity in the same framework as other forces.

Gravitons (quantized gravitational waves) could be the key to linking the smooth geometry of spacetime (general relativity) with the discrete, probabilistic nature of quantum fields (QFT).
String theory and loop quantum gravity are leading candidates in the search for a unified framework, with string theory providing a quantum description of spacetime via vibrating strings, and LQG offering a picture of spacetime as a network of quantized loops.

In summary, spacetime oscillations as quantum fields is not only a valid path but one of the core ideas in many modern approaches to quantum gravity. While we do not yet have a complete theory, this perspective plays a central role in the ongoing effort to reconcile the two most successful theories of modern physics.

Can we reach reconciliation among QFT and GR by finding formalism of length and time in Planck scale into Ricci geometri in GR?

Yes, reaching a reconciliation between Quantum Field Theory (QFT) and General Relativity (GR) might indeed be possible by finding a formalism that incorporates Planck-scale physics into the framework of Ricci geometry in GR. This would involve quantizing spacetime itself and understanding how the classical geometry of GR, which operates at macroscopic scales, transitions to a quantum description at the Planck scale. The Planck scale is where quantum effects of gravity are expected to dominate, and spacetime is no longer smooth but exhibits quantum fluctuations.

Key Ideas in Reconciling QFT and GR at the Planck Scale:

1. Planck Scale: The Quantum Gravity Regime: The Planck length ( meters) and Planck time ( seconds) represent the scales where the effects of quantum gravity become significant. At these scales, the usual classical description of spacetime in terms of smooth Ricci geometry breaks down, and quantum fluctuations in spacetime need to be taken into account.

2. Ricci Geometry in General Relativity: In GR, spacetime is described by Ricci geometry, a continuous, differentiable manifold with curvature determined by the Einstein field equations:

R_{\mu\nu} - \frac{1}{2}g_{\mu\nu}R + \Lambda g_{\mu\nu} = 8\pi G T_{\mu\nu}

How Can Planck-Scale Physics Inform Ricci Geometry?

To reconcile QFT and GR, we need to extend the formalism of spacetime geometry to account for quantum properties of spacetime at the Planck scale. This might involve modifying Ricci geometry to include quantized lengths, times, and possibly even a discrete structure of spacetime.

Potential Approaches:

1. Effective Field Theory and Quantum Corrections to GR: One approach is to treat general relativity as an effective field theory that works well at large scales but requires quantum corrections at smaller (Planck) scales. In this framework:

At scales much larger than the Planck length, spacetime appears smooth, and the classical Einstein field equations hold.

As we approach the Planck scale, quantum fluctuations in spacetime geometry would introduce corrections to the Einstein equations, modifying the behavior of spacetime curvature (Ricci tensor) and the metric.

These corrections could be described using higher-order terms in the curvature (such as  or  terms) and possibly other quantum effects. This is an approach used in quantum gravity as an effective field theory, where the leading-order behavior is classical, and quantum corrections become important at very small scales.

2. Quantization of Spacetime: Discretizing Geometry: At the Planck scale, spacetime may not be smooth and continuous as in Ricci geometry but instead take on a discrete, quantum structure. This is the focus of approaches such as loop quantum gravity (LQG):

Loop Quantum Gravity proposes that spacetime is composed of discrete "chunks" or quanta of space. These are described mathematically as spin networks, and the areas and volumes of spacetime are quantized.

The challenge is to link this discrete quantum geometry with the smooth, continuous geometry of GR at larger scales. The quantum states of geometry in LQG would recover classical Ricci geometry in the limit of large distances (much larger than the Planck scale), but at the smallest scales, spacetime would behave very differently.

In this framework, length and time at the Planck scale are no longer continuous variables but take on discrete values, which would fundamentally alter how we think about spacetime geometry.

3. String Theory and the Holographic Principle: String theory offers another approach to reconciling QFT and GR by introducing a new formalism for spacetime:
In string theory, all particles (including the graviton, which mediates gravity) are viewed as different vibrational modes of fundamental strings.

At the Planck scale, the concept of spacetime might emerge from the interactions of these strings, which could provide a quantum description of gravity. This would potentially modify Ricci geometry to account for quantum effects at the Planck scale.
The holographic principle, derived from string theory and black hole physics, also suggests that the information content of a region of spacetime can be described by quantum fields on the boundary of that region. This could provide a bridge between the quantum field theoretic description of spacetime and its geometric description in GR.

4. Asymptotic Safety and Renormalization of Gravity: Another approach is the asymptotic safety scenario, which proposes that gravity becomes renormalizable at very high energies (near the Planck scale). In this scenario, there is a non-trivial fixed point in the renormalization group flow of gravity, where quantum effects become important and modify the classical behavior of spacetime.

In this framework, length and time at the Planck scale would be influenced by quantum corrections, and the behavior of spacetime curvature (Ricci tensor) would change accordingly. The Ricci geometry might still apply but with quantum modifications at small scales, where spacetime no longer behaves classically.

5. Non-Commutative Geometry: Another speculative approach is non-commutative geometry, which modifies the standard notion of spacetime coordinates by introducing uncertainty relations similar to those in quantum mechanics. 

In this framework:

The coordinates of spacetime do not commute, meaning that there is an inherent uncertainty in measurements of spacetime position at very small scales (similar to the uncertainty principle for position and momentum in quantum mechanics).

This leads to a modification of the usual spacetime geometry, potentially reconciling QFT with GR by introducing quantum uncertainty into the fabric of spacetime itself.

Conclusion: A Path to Reconciliation

To reconcile QFT and GR, finding a formalism for length and time at the Planck scale within the framework of Ricci geometry is a promising approach. Several approaches to quantum gravity seek to modify or extend the classical Ricci geometry of general relativity to account for quantum effects at small scales:

Effective field theory approaches can introduce quantum corrections to classical GR.

Loop quantum gravity and discrete spacetime models suggest that spacetime itself is quantized at the Planck scale, and Ricci geometry emerges from a more fundamental quantum description.
String theory and the holographic principle propose new ways to think about spacetime, where it is an emergent concept from more fundamental quantum fields.

In all these cases, the goal is to extend the classical geometry of GR (Ricci tensor, metric) to a quantum regime where spacetime is no longer smooth and continuous but instead exhibits quantum fluctuations and discrete structures at the Planck scale. This approach could ultimately lead to a unified theory of quantum gravity, reconciling the successes of both QFT and GR.

Consider these some points:1. Energy of photon E = hf. 2. Photon interacts with gravity. 3. Mass energy equivalent equation  E = mc^2. When we can find mathematic bridge by formalism f = m in which derived from hf = mc^2, is it alternative way to reconcile GR = QFT?

Yes, exploring the formalism where , derived from , could be an interesting alternative approach to reconciling General Relativity (GR) and Quantum Field Theory (QFT). Let's walk through the physical meaning of this relationship and how it might help bridge the gap between these two foundational theories of physics.

1. Photon Energy and Frequency:
In quantum mechanics, the energy of a photon is given by Planck’s relation:
E = hf
h is Planck’s constant,
f is the frequency of the photon.
This equation connects a quantum property (frequency ) with energy, and it is central to quantum mechanics and quantum field theory (QFT). It emphasizes the particle-wave duality of photons, where their energy is directly tied to their frequency.

2. Mass-Energy Equivalence:
In Einstein’s special theory of relativity, the famous mass-energy equivalence formula is:
E = mc^2
m is the rest mass,
c is the speed of light.
This equation connects mass to energy and highlights that mass is a form of energy. This is crucial for general relativity (GR), where the presence of energy (including mass) causes spacetime to curve, as described by the Einstein field equations.

3. Photon Interaction with Gravity (General Relativity)

Even though photons are massless, they interact with gravity because they carry energy and momentum, which, according to GR, still contribute to spacetime curvature. This is why light is bent by gravity (gravitational lensing) and why photons can lose energy (redshift) in the presence of a strong gravitational field.
In this context, both energy and mass influence spacetime curvature, meaning photons, despite being massless, still feel the effects of gravity.

4. The Formalism

Let’s now combine the equations:

hf = mc^2
f = \frac{mc^2}{h}

What Could  Imply?

The equation  (derived from ) conceptually implies that the frequency of a quantum object (like a photon) can be related to an effective mass via the equivalence of energy and mass.

5. Reconciling GR and QFT Through
This relation can offer several possible insights into bridging GR and QFT:

(a) Quantum Interpretation of Gravity:
In GR, gravity is sourced by mass and energy, which curve spacetime. If we treat frequency as being equivalent to mass via , this implies that quantum properties (like the frequency of a photon) can have a gravitational effect, even though we usually think of mass as the gravitational source.

This formalism hints at the idea that gravitational effects can be linked to quantum frequencies, which might help us develop a quantum theory of gravity. In such a theory, not only mass but also the quantum properties (like frequency and energy) of particles would influence the curvature of spacetime.

(b) Photon as a Bridge Between Mass and Energy:
Photons, despite being massless, carry energy through their frequency. The relation  can be viewed as a way of treating the photon’s frequency as a type of "effective mass" in the context of gravitational interactions.

If we interpret frequency as mass, then the bending of light (photon interaction with gravity) could be seen as a manifestation of the photon's "effective mass" curving spacetime in the same way that normal mass does. This interpretation could provide a way to understand gravity at the quantum level, where particles do not have mass in the traditional sense, but still carry energy and frequency.

(c) Energy and Geometry:

The relation  could potentially lead to a new way of thinking about spacetime curvature. In GR, spacetime is curved by the presence of mass and energy. If frequency (a quantum property) is analogous to mass, this could imply that quantum properties, like frequency, curve spacetime in the same way mass does.

This might open up the possibility of describing spacetime geometry using quantum field theory language, where the curvature of spacetime is influenced not just by classical mass-energy, but by the quantum fields (with associated frequencies) themselves.

6. Implications for Quantum Gravity:
The formalism  hints at a duality between mass-energy in classical GR and quantum properties (frequency) in QFT. This duality could potentially be developed into a consistent framework for quantum gravity, where:

Mass and frequency (or energy) are treated as equivalent sources of gravitational interaction.
Spacetime curvature in GR could be linked to quantum properties (like frequency) through some quantum version of the Einstein field equations.

One potential avenue could be formulating a quantum theory of gravity where the quantum fields themselves (described by their frequency or energy) directly affect spacetime curvature. This is analogous to how in QFT, quantum fields govern the interactions of particles, but now extended to include gravitational interactions.

7. Limitations and Challenges:
While this formalism provides a conceptual bridge between mass and frequency, there are challenges:

Quantum Gravity Formalism: Developing a full quantum theory of gravity remains an open problem. Current approaches like loop quantum gravity (LQG) and string theory aim to describe quantum gravity but are not yet complete or fully tested.

Non-massive Particles in GR: GR primarily deals with massive objects curving spacetime. Extending this to massless particles (photons) via frequency requires modifications to classical GR or developing a new quantum version of Einstein’s equations.

Conclusion:

The formalism , leading to , offers an interesting conceptual bridge between QFT and GR by relating quantum properties (frequency) to classical mass-energy equivalence. This could provide an alternative way to reconcile the two theories by suggesting that quantum fields and their properties (like frequency) directly influence spacetime curvature, much like mass and energy do in general relativity.

This formalism might lead to the development of a quantum version of general relativity, where quantum fields play a central role in shaping the geometry of spacetime. While still speculative, it offers a promising avenue for exploring how quantum and gravitational phenomena could be unified into a single, consistent framework.
Some variables in entanglement are identical with variables of wormhole, so we can write equation 

ER = EPR, how can we create formalism from this to reconcile GR and QFT?

The conjecture that ER = EPR, proposed by physicists Juan Maldacena and Leonard Susskind, suggests a deep connection between Einstein-Rosen bridges (wormholes, or ER bridges) and Einstein-Podolsky-Rosen (EPR) pairs of entangled particles. In essence, this conjecture posits that quantum entanglement between two particles is somehow related to the existence of a wormhole connecting two regions of spacetime.

This idea is a bold step toward reconciling General Relativity (GR), which describes gravity and spacetime curvature, and Quantum Field Theory (QFT), which describes particles and quantum interactions. The core idea of ER = EPR is that the phenomenon of quantum entanglement (which is central to QFT) and wormholes (which are solutions of the Einstein field equations in GR) are two manifestations of the same underlying physics.

Breaking Down ER = EPR

1. ER: Einstein-Rosen Bridges (Wormholes):
In GR, an Einstein-Rosen bridge is a solution of the Einstein field equations that describes a wormhole: a shortcut through spacetime connecting two distant points.
Wormholes can be thought of as non-traversable (in most physical contexts) and are often associated with black holes. They describe a deep connection between different parts of spacetime, and they are a purely geometric solution in GR.

2. EPR: Einstein-Podolsky-Rosen Entanglement:
In QFT and quantum mechanics, EPR pairs refer to pairs of entangled particles that are correlated in such a way that the state of one particle instantaneously affects the state of the other, no matter how far apart they are. This nonlocal connection is central to quantum mechanics and does not involve any classical spacetime bridge, like a wormhole.

This form of quantum entanglement demonstrates nonlocality, where information about one particle's state seems to "travel" instantaneously to the other, violating classical intuitions of spacetime separation.

ER = EPR: The Connection

The ER = EPR conjecture posits that these two seemingly different phenomena—wormholes (ER) in GR and entanglement (EPR) in QFT—are actually different manifestations of the same underlying reality. Specifically:

Entangled particles might be connected by microscopic wormholes.

These wormholes are non-traversable but still provide a geometric description of the entanglement between two distant particles.
This idea provides a conceptual framework that might help reconcile GR and QFT. To make this conjecture rigorous, we would need a formalism that describes both entanglement and wormholes in the same mathematical language. Below are steps and potential approaches toward achieving that goal.

Steps Toward a Formalism to Reconcile GR and QFT via ER = EPR

1. Duality Between Geometry (GR) and Quantum Information (QFT): The essence of the ER = EPR conjecture lies in a duality between spacetime geometry and quantum information. To build a formalism around this, we need to express both quantum entanglement and wormholes in a common framework. Some key areas that could lead to this are:

Holography and AdS/CFT Correspondence: The AdS/CFT correspondence is one of the most promising tools for connecting GR and QFT. It proposes that a theory of quantum gravity in a higher-dimensional spacetime (anti-de Sitter space, AdS) is equivalent to a conformal field theory (CFT) on the boundary of that spacetime. 

In this framework:

The geometry of spacetime in the bulk (including wormholes) corresponds to quantum states on the boundary (including entangled states like EPR pairs).

This duality suggests that spacetime geometry and quantum entanglement are deeply related, providing a mathematical bridge between GR (which governs the bulk) and QFT (which governs the boundary).

ER = EPR might be understood as a holographic duality in AdS/CFT, where entanglement (EPR pairs) is reflected in the bulk as connected wormholes (ER bridges).

2. Geometrizing Quantum Information: To formalize ER = EPR, we need to geometrize quantum information, meaning we need to describe quantum entanglement (which is an abstract quantum phenomenon) in terms of spacetime geometry. This could be done through:

Quantum Error Correction and Tensor Networks: 

Quantum error correction codes and tensor network models have been used to describe the entanglement structure of quantum systems and how it might correspond to spacetime geometry. 

In this view:

Quantum states of entangled particles can be mapped onto networks of quantum gates or tensors, which in turn can be mapped onto a spacetime geometry.

Tensor networks like MERA (Multiscale Entanglement Renormalization Ansatz) suggest a holographic interpretation of quantum states, where the geometry of spacetime (like a wormhole) emerges from entanglement patterns in quantum systems.

If we can rigorously formalize this idea, we could connect the entanglement entropy of quantum states to geometric quantities like the area of wormhole mouths or the volume of ER bridges. This would provide a mathematical tool to bridge GR and QFT.

3. Entanglement Entropy and Spacetime Emergence: Entanglement entropy measures how entangled a system is, and in some cases (such as in black hole physics), it is related to the area of the event horizon. This suggests a connection between quantum information and the geometry of spacetime.

Ryu-Takayanagi Formula: In the AdS/CFT correspondence, the Ryu-Takayanagi formula connects the entanglement entropy of a quantum system to the area of a minimal surface in the bulk spacetime. This indicates that spacetime geometry is encoded in the quantum entanglement of the boundary theory.

If ER = EPR is true, we might extend this idea to say that entanglement between quantum particles (EPR pairs) gives rise to wormhole-like connections in spacetime. By finding a formalism that connects the entropy of entanglement with the geometry of wormholes, we could develop a framework that unifies quantum information (QFT) with spacetime geometry (GR).

4. Non-Traversable Wormholes as a Model for Entanglement:

Non-traversable wormholes, which are solutions to the Einstein field equations, do not allow physical objects to travel through them, but they can still represent a connection between two regions of spacetime. These are analogous to quantum entanglement, where information about one particle’s state is instantaneously connected to the other, but no information can travel faster than light.

To formalize ER = EPR, we need a mathematical framework where non-traversable wormholes correspond to quantum correlations (entanglement) between distant particles. This would require a modification of Einstein's equations to account for quantum degrees of freedom (like entanglement) as a source of spacetime geometry.

Possible Formalism for ER = EPR

1. Einstein's Equations with Quantum Entanglement Terms: We could modify Einstein’s field equations to include terms that account for quantum entanglement as a source of spacetime curvature. This would involve coupling the classical stress-energy tensor  with a term that represents quantum information or entanglement.

The modified Einstein equations might take the form:

R_{\mu\nu} - \frac{1}{2}g_{\mu\nu}R + \Lambda g_{\mu\nu} = 8\pi G \left( T_{\mu\nu} + T_{\mu\nu}^{\text{quantum}} \right)

2. Quantum Circuit Representation of Spacetime: Another formalism might involve representing spacetime as a quantum circuit, where each EPR pair is represented by a quantum gate in a tensor network, and wormholes correspond to specific connections between these gates. This could be realized through a discrete model of spacetime, where each quantum bit (qubit) represents a piece of spacetime, and the entanglement structure dictates the geometry.

3. Entanglement as the Fundamental Building Block of Spacetime: The ER = EPR conjecture suggests that spacetime might emerge from the entanglement structure of quantum fields. A formalism based on entanglement entropy and its relationship to spacetime curvature could serve as a bridge between GR and QFT, where spacetime geometry is reconstructed from quantum entanglement

Conclusion

The ER = EPR conjecture presents a profound insight: that wormholes in general relativity and quantum entanglement in QFT might be two sides of the same coin. To reconcile GR and QFT through this idea, we need to develop a formalism that:

1. Describes quantum entanglement as a geometric feature of spacetime.

2. Modifies Einstein’s equations to include quantum information terms, such as entanglement entropy or other quantum correlation measures, which can serve as sources of spacetime curvature.

3. Explores the holographic principle (as in AdS/CFT), where spacetime geometry and quantum information are dual descriptions of the same physical reality, potentially using the Ryu-Takayanagi formula to relate entanglement to spacetime surfaces.

4. Develops a framework where non-traversable wormholes are interpreted as the geometric representation of quantum entanglement, thus linking classical geometry (wormholes) with quantum phenomena (EPR pairs).

To summarize, formalizing ER = EPR requires bridging the conceptual gap between quantum information theory and spacetime geometry. By connecting quantum entanglement with wormhole structures, we might uncover a new perspective on how spacetime and quantum fields interact. This could eventually lead to a unified description of gravity and quantum mechanics, providing a quantum theory of gravity that integrates both General Relativity and Quantum Field Theory into a coherent framework.

Penutup

Kelima opsi dan kemungkinan di atas, walaupun tampak logis, membuat formalisme matematisnya tidaklah mudah, bahkan sangat sulit. Sejak Newton, Maxwell, dan Einstein semua kerangka kerja fisika harus menggunakan formalisme matematis. Eksperimen pikiran dalam fisika bukanlah filsafat, tapi gerbang kepada formalisme matematis.

Bacaan Lanjutan 

https://phys.org/news/2024-10-planck-scale-dark-leveraging-quantum.html