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I'm trying to understand from where the first particles in quantum mechanics came from, and while doing research I got this information from an article about the origin of matter:

In the beginning, there was not yet any matter. However, there was a lot of energy in the form of light, which comes in discrete packets called photons. When photons have enough energy, they can spontaneously decay into a particle and an antiparticle

Many of the bosons around just after the big bang were so energetic that they could decay into much more massive particles such as protons (remember, $E=mc^2$, so to make a particle with a large mass m, you need a boson with a high energy $E$). The mass in the universe came from such decays.

According to Einstein's theory of special relativity $E=mc^{2}$, the mass of a body is a measure of its energy-content - Yet, I don't understand how a matter quantity increases in terms of energy.

Also I found this information on Wikipedia about photons' matter creation:

Because of momentum conservation laws, the creation of a pair of fermions (matter particles) out of a single photon cannot occur. However, matter creation is allowed by these laws when in the presence of another particle (another boson, or even a fermion) which can share the primary photon's momentum. Thus, matter can be created out of two photons.

Let's assume there was an infinite energy within an infinite time interval. Where did the first photon, boson or even fermion came from, were they born from just energy? How can the infinite universe mass, come from nothing?

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The article you quote is outdated, to say the least:

However, there was a lot of energy in the form of light, which comes in discrete packets called photons. When photons have enough energy, they can spontaneously decay into a particle and an antiparticle

Photons do not decay, they have to interact with other particles/fields in order to generate particle antiparticle pairs, so this is not correct in the quote too.

I would advise you to read further the links I provide here for a true state of the model of the creation of the present day universe.

It is not photons that carry primordial energy. In the current Big Bang model, there was a lot of energy in the inflation period, but the particles that were carrying it were the inflatons:

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The inflaton field is a hypothetical scalar field that is theorized to drive cosmic inflation in the very early universe. The field, originally theorized by Alan Guth, provides a mechanism by which a period of rapid expansion from $10^{−35}$ to $10^{−34}$seconds after the initial expansion can be generated, forming a universe consistent with observed spatial isotropy and homogeneity.

The research is directed into modeling mathematically the time evolution from the hypothetical inflaton particle to the energy being carried by the particles we have measured and modeled mathematically into the standard model of particle physics.

Yet, I don't understand how a matter quantity increases in terms of energy.

This means you have not studied the theory of special relativity. In special relativity energy and momentum are connected into a four vector , whose "length" is the mass of the particle described by that four vector, which is invariant in Lorenz transformations. So the masses in the particle table are invariant . Their energy is a function both of the rest mass energy and the kinetic energy as can be seen in the link. So in special relativity mass and energy can be transformed in particular interactions, as has been measured in nuclear processes and particle interactions.

Let's assume there was an infinite energy within an infinite time interval. Where did the first photon, boson or even fermion came from, were they born from just energy?

How can the infinite universe mass, come from nothing?

For people really interested I would advice reading this blog post by Motl (a high reputation contributor here) where he explains how energy conservation has been modeled mathematically, and how this holds for special relativity, but not for general relativity. Energy conservation works for flat spaces, but not for high curvatures of space time, which is what happened at the beginning of the universe when the energy for all the matter and energy we see, was generated. He concludes:

The main lesson here is that general relativity is not a theory that requires physical objects or fields to propagate in a pre-existing translationally invariant spacetime. That's why the corresponding energy conservation law justified by Noether's argument either fails, or becomes approximate, or becomes vacuous, or survives exclusively in spacetimes that preserve their "special relativistic" structure at infinity. At any rate, the status of energy conservation changes when you switch from special relativity to general relativity.

The result is that energy at those times can appear out of nothing, i.e. we are mathematically allowed to model the beginning of the universe with energy which appears out of nothing.

The title, in contrast to the content, of your question asks an existential question.

Are particles an abstraction of energy, or their matters do indeed exist?

We define matter macroscopically, as the coffee we drink and the chair we sit on. In classical physics which developed from measurements and observations, mass is a conserved quantity. Then we discovered radioactivity and the whole micro world of quantum mechanics and had to redefine mass and energy exchanges. At the overlap classical emerges from the underlying quantum mechanical background , i.e. macroscopically mass is a conserved quantity for most everyday cases. For high velocities, small dimensions, and radioactivity one has to define "entities" which behave as particles with a fixed mass in interactions but have probabilistic distributions that show interference patterns of waves, as this interference pattern appearing in single electron double slit scatters.

So then one needs a definition of "exist" which is outside of physics discussions.

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